Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 49 (2008), No. 2, 491-515.

On the Structure of Convex Sets with Applications to the Moduli of Spherical

Minimal Immersions

Gabor Toth

Department of Mathematics, Rutgers University Camden, New Jersey 08102 e-mail: gtoth@crab.rutgers.edu

Abstract. We study the properties of certain affine invariant measures of symmetry associated to a compact convex body L in a Euclidean vector space. As functions of the interior of L, these measures of symmetry are proved or disproved to be concave in specific situations, notably for the reduced moduli of spherical minimal immersions. MSC 2000: 53C42 Keywords: convex set, distortion, measures of symmetry

1. Introduction and statement of results

Let E be a Euclidean vector space of dimension n. If K ⊂ E then [K] and ⟨K⟩ denote the convex hull and the affine span of K, respectively. Since [K] has a nonempty relative interior in ⟨K⟩, it is a convex body in ⟨K⟩. Let L be a compact convex body in E and O a point in the interior of L. As in [9, 10], we define an affine invariant σ(L, O) as follows. Given C ∈ ∂L, the line passing through O and C intersects ∂L in another point that we call the opposite of C with respect to O and denote it by C^{o}. Clearly, (C^{o})^{o} = C. We define the distortion function Λ : ∂L × int L → R as

Λ_{L}(C, O) = Λ(C, O) = \frac{d(C, O)}{d(C^{o}, O)}, C ∈ ∂L, O ∈ int L,

0138-4821/93 $ 2.50 \begin{matrix}c \\ ◯ \end{matrix} 2008 Heldermann Verlag

492 G. Toth: On the Structure of Convex Sets with Applications …

where d(X, X′) = |X − X′| is the Euclidean distance between the points X and X′ in E, and C^{o} is the opposite of C with respect to O. The distortion Λ is a continuous function [3]. By definition, Λ(C^{o}, O) = 1/Λ(C, O). The minimum distortion λ(O) = inf_{C∈∂L} Λ(C, O), as a function of the interior of L has an extensive literature. (We refer to the survey article of Grünbaum [5] and the references therein.) In particular, the structure of the level sets and the critical set of Minkowski’s measure of symmetry sup_{O∈ int L} λ(O) ≥ 1/n have been studied by many authors. The derived measures λ(O_{0}), where O_{0} is a specific center of L also have an extensive literature. Choices of O_{0} include the centroid, the centers of circumscribed and inscribed ellipsoids, the centroid of the surface area of ∂L, and the curvature centroid. A multi-set {C_{0}, …, C_{n}} is called a configuration with respect to O if {C_{0}, …, C_{n}} ⊂ ∂L and O ∈ [C_{0}, …, C_{n}]. Let C(L, O) denote the set of all configurations of L. We define

σ(L, O) = inf_{\{C_{0},...,C_{n}\}∈C(L,O)} ∑_{i=0}^{n} \frac{1}{1 + Λ(C_{i}, O)}.  (1)

A configuration {C_{0}, …, C_{n}} for which the infimum is attained is called minimal. Minimal configuration exists since L is compact. σ(L, O) is a continuous function on the space of compact convex bodies with specified interior point, and it is also invariant under affine transformations. In addition, by [10], 1 ≤ σ(L, O) ≤ (n + 1)/2. The lower bound is attained by simplices, and the upper bound is realized by symmetric L (with respect to O). Because of these properties, σ(L, O) is a measure of symmetry in the sense of Grünbaum [5]. σ(L, O) is related to the measures of symmetry

inf_{{C_{0},...,C_{n}}∈C(L,O)} ∑_{i=0}^{n} Λ(C_{i}, O) and inf_{{C_{0},...,C_{n}}∈C(L,O)} ∏_{i=0}^{n} Λ(C_{i}, O)

for which upper and lower bounds have been derived. (See again Grünbaum [5].)

According to [9] (Theorem D), σ(L, .) : int L → R is continuous and has the property

lim_{d(O,∂L)→0} σ(L, O) = 1.

In particular, it extends continuously to the boundary of L with value 1. A point O ∈ int L is said to be regular if, for any minimizing configuration {C_{0}, …, C_{n}} ∈ C(L, O), the convex hull [C_{0}, …, C_{n}] is a simplex which contains O in its interior. The set R of regular points is called the regular set of L. This is an open (possibly empty) subset of int L. If O is a regular point then, at each point C in a minimal configuration, Λ(., O) assumes a local maximum. Therefore, there exists a pair of parallel supporting hyperplanes passing through C and C^{o}. (This follows from the description of ∂L near C in Section 7 of [9].) Thus, the segment [C, C^{o}] is an affine diameter of L. We obtain that if O is a regular point then there are n + 1 (affinely) independent affine diameters of L passing through O. Note that it is a difficult and unsolved problem to characterize those points in the interior of L at which n + 1 affine diameters pass through. (See Grünbaum [5] and Koziński [6, 7].)

In [9] (Proposition 2 and Theorem E), we proved the following:

Theorem A. The function σ(L, .) is concave on the regular set R. For n = 2, it is concave on the entire L.

One of the main purposes of this paper is to see how far can this result be extended to specific classes of compact convex bodies. The main technical result is to calculate σ(L, .) for any convex cone L in terms of certain affine invariants on the base of the cone. This will give the following:

Theorem B. For any 3-dimensional compact convex cone L, the function σ(L, .) is concave on int L. There exists a 4-dimensional compact convex cone L such that σ(L, .) is not concave on int L.

The key to prove Theorem B is to extend σ to a sequence of invariants {σ_{m}}_{m≥1}, with σ_{n} = σ, and study the monotonicity properties of this sequence. Let m ∈ N. A multi-set {C_{0}, …, C_{m}} is an m-configuration with respect to O if {C_{0}, …, C_{m}} ⊂ ∂L and O ∈ [C_{0}, …, C_{m}]. For an m-configuration {C_{0}, …, C_{m}} the convex hull [C_{0}, …, C_{m}] is a convex polytope in its affine span ⟨C_{0}, …, C_{m}⟩. This polytope has maximum dimension m iff it is a simplex. In this case, we call {C_{0}, …, C_{m}} simplicial. We define

σ_{m}(L, O) = inf_{\{C_{0},...,C_{m}\}∈C_{m}(L,O)} ∑_{i=0}^{m} \frac{1}{1 + Λ(C_{i}, O)},

where C_{m}(L, O) denotes the set of all m-configurations of L. An m-configuration for which the infimum is attained is minimal. Compactness implies that minimal configurations exist. σ_{m}(L, .) : int L → R is continuous [9] (Theorem D), and extends continuously to ∂L to 1 since

lim_{d(O,∂L)→0} σ_{m}(L, O) = 1.  (2)

Since a 1-configuration of L is an opposite pair of points {C, C^{o}} ⊂ ∂L, we have σ_{1}(L, O) = 1. In what follows, we suppress O when no confusion arises. By [9] (Theorem B), for m ∈ N, we have

1 ≤ σ_{m}(L) ≤ \frac{m + 1}{2}.  (3)

If σ_{m}(L) = 1 then m ≤ n and there exists an affine subspace F ⊂ E, O ∈ F, of dimension m such that L∩F is an m-simplex. In this case a minimal configuration {C_{0}, …, C_{m}} ∈ C_{m}(L ∩ F) is simplicial. Moreover, we have

∑_{i=0}^{m} \frac{1}{1 + Λ(C_{i})} = 1 and ∑_{i=0}^{m} \frac{1}{1 + Λ(C_{i})}C_{i} = O.  (4)

Conversely, if L has a simplicial intersection with an m-dimensional affine subspace F ∋ O then σ_{m}(L) = 1. For m ≥ 2, σ_{m}(L) = (m + 1)/2 if and only if Λ = 1 on ∂L, that is, if and only if L is symmetric. As before, σ_{m}(L, O), m ≥ 1, are measures of symmetry. In general, for m′ ≥ m, we obviously have

σ_{m′}(L) ≤ σ_{m}(L) + \frac{m′ − m}{1 + max_{∂L} Λ}.  (5)

For m ≥ n, n = dim L, equality holds in (5) [10]:

σ_{m}(L) = σ_{n}(L) + \frac{m − n}{1 + max_{∂L} Λ}.

Equivalently, the sequence {σ_{m}(L)}_{m≥1} is arithmetic (with difference 1/(1 + max_{∂L} Λ)) from the n-th term onwards. It is also clear that, for m < n, we have

σ_{m}(L) = inf_{O∈F⊂E, dim F=m} σ(L ∩ F),

where the infimum is over affine subspaces F ⊂ E. In [9] (Theorem B) the following superadditivity was proved:

σ_{m+m′}(L) − σ_{m+1}(L) ≥ σ_{m′}(L) − σ_{1}(L), m ≥ 0, m′ ≥ 2.

In particular, setting m′ = 2 and using σ_{2}(L) ≥ 1, we see that the sequence {σ_{m}(L)}_{m≥1}, is nondecreasing. In Section 2 we will derive a stronger statement, namely, that, after a possible string of 1’s, this sequence is strictly increasing. The original motivation for the measures of symmetry {σ_{m}(L)}_{m≥1} is the bulging phenomenon observed in moduli spaces of spherical minimal immersions [11, 13]. The general setting for the moduli is as follows [1, 2, 11]. Let H be a Euclidean vector space and S_{0}^{2}(H) the space of tracefree symmetric endomorphisms of H. We define the reduced moduli space by

K_{0} = K_{0}(H) = {C ∈ S_{0}^{2}(H) | C + I ≥ 0},

where ≥ 0 means positive semi-definite. Then K_{0} is a compact convex body in S_{0}^{2}(H). The distortion Λ(C) (with respect to the origin) of an endomorphism C ∈ ∂K_{0} is the maximal eigenvalue of C. Finally, the moduli space corresponding to a linear subspace E ⊂ S_{0}^{2}(H) is the intersection L = E ∩ K_{0}. It is a difficult and important problem to describe the geometry of L. More specifically, let M be a compact (isotropy irreducible) Riemannian homogeneous manifold, λ an eigenvalue of the Laplace-Beltrami operator acting on functions of M, and H the eigenspace of functions corresponding to λ. Then the DoCarmo-Wallach moduli space of spherical minimal immersions of M is the moduli space of K_{0} = K_{0}(H) corresponding to a specific choice E of a linear subspace in S_{0}^{2}(H). For details, see [11].

Theorem C. We have σ_{1}(K_{0}, .) = ⋯ = σ_{h−1}(K_{0}, .) = 1, h = dim H. For m ≥ h, we have

σ_{m}(K_{0}, O) = inf_{\{C_{0},...,C_{m}\}∈C_{m}(K_{0})} ∑_{i=0}^{m} \frac{1 + ⟨O, C_{i}⟩/h}{1 + Λ(C_{i})},

where C_{m}(K_{0}) is the set of m-configurations of K_{0} with respect to the origin. Consequently, σ_{m}(K_{0}, .) is concave on int K_{0} and attains its unique maximum (m+1)/h at the origin.

The structure of a moduli space L = E ∩ K_{0} is much more subtle. Even the first statement of Theorem C leads to an unsolved problem. Let r(L) be the largest positive integer such that σ_{m}(L) = 1 for 1 ≤ m ≤ r(L). We have r(L) ≤ dim K − n(L), where n(L) = min { rank (C + I) | C ∈ ∂L}. In the setting of spherical minimal immersions, n(L) − 1 is the minimal range dimension for such immersions. To determine n(L) is an old and difficult problem [1], [2], [8], [11], [12], [13].

Corollary. Let M be a compact isotropy irreducible Riemannian homogeneous space, λ an eigenvalue of the Laplace-Beltrami operator on M, H the corresponding eigenspace, and L the moduli space of spherical minimal immersions of M. Assume that the sequence {σ_{m}(L)}_{m≥1} starts with a string of 1’s of length r. Then there exists a spherical minimal immersion of M into a sphere of dimension < dim H − r.

Acknowledgment. The author wishes to thank the referee for the careful reading and suggestions which led to the improvement of the original manuscript.

2. The measures σ(L, O) and σ_{n−k}(L, O), 0 ≤ k < n

In view of (5) we define the regular set R ⊂ int L as

R = \{O ∈ int L | σ(L, O) < σ_{n−1}(L, O) + \frac{1}{1 + max_{∂L} Λ(., O)}\} .

By continuity of the functions in the defining inequality of R, the set R is open in int L and hence in E. Again by (5), for 1 ≤ k < n, we define

S_{k} = \{O ∈ int L | σ(L, O) = σ_{n−k}(L, O) + \frac{k}{1 + max_{∂L} Λ(., O)}\} .

We call S = S_{1} the singular set. S is the complement of R in int L and therefore it is relatively closed in int L. Clearly, we have

S_{k+1} ⊂ S_{k}, k = 1, …, n − 2.

Let O ∈ S. We define the degree of singularity of O as the largest k such that O ∈ S_{k}. By the above, the degree of singularity of O is n − m if and only if the sequence {σ_{m}(L, O)}_{m≥1} is arithmetic (with difference 1/(1 + max_{∂L} Λ(., O))) exactly from the m-th term onwards.

Proposition 1. O ∈ R if and only if every minimal configuration is simplicial and O is contained in the interior of the corresponding n-simplex. In this case, Λ(., O) attains its local maximum at each configuration point. Let O ∈ S. Then O ∈ S_{n−m} if and only if there exists a minimal n-configuration which contains an m-configuration. In this case, the m-configuration is also minimal, and at each n-configuration point complementary to the m-configuration Λ(., O) attains absolute maximum. If, in addition, the degree of singularity of O is n − m (equivalently, O ∉ S_{n−m+1}), then the m-configuration is simplicial, O is contained in the relative interior of the corresponding m-simplex, and Λ(., O) restricted to the affine span of the m-configuration attains local maxima at every m-configuration point.

Proof. We first observe that any subconfiguration of a minimal configuration is also minimal, and the distortion attains absolute maximum at the complementary points in the respective affine span. Now let 1 ≤ m ≤ n−1. We claim that O ∈ S_{n−m} if and only if there exists a minimal n-configuration {C_{0}, …, C_{n}} ∈ C_{n}(L, O) which contains an m-configuration {C_{0}, …, C_{m}} ∈ C_{m}(L, O), where we relabeled the configuration points if necessary. The entire proposition follows from this claim. Indeed, setting m = n − 1, the claim implies that O ∈ R if and only if every minimal n-configuration {C_{0}, …, C_{n}} ∈ C_{n}(L, O) contains no subconfiguration, or equivalently, the corresponding convex hull [C_{0}, …, C_{n}] is an n-simplex and O is not on its boundary. Thus the claim implies the first statement of the proposition. The degree of singularity of O is n − m if and only if m is the least number in the claim. Indeed, if the subconfiguration {C_{0}, …, C_{m}} ∈ C_{m}(L, O) is not simplicial or O is on the boundary of its convex hull then this contradicts the minimality of m. Thus, the claim implies the remaining part of the proposition. By the observation above, we need to prove the “only if” part of the claim. Let O ∈ S_{n−m}. Choose a minimal m-configuration {C_{0}, …, C_{m}} ∈ C_{m}(L, O) and extend it to an n-configuration {C_{0}, …, C_{n}} ∈ C_{n}(L, O) by adding C_{m+1} = ⋯ = C_{n} at which Λ(., O) attains absolute maximum. With this, we have

∑_{i=0}^{n} \frac{1}{1 + Λ(C_{i})} = ∑_{i=0}^{m} \frac{1}{1 + Λ(C_{i})} + \frac{n − m}{1 + max_{∂L} Λ}
= σ_{m}(L, O) + \frac{n − m}{1 + max_{∂L} Λ}
= σ(L, O).

Thus, {C_{0}, …, C_{n}} is minimal and the claim follows.

Proposition 2. The sequence {σ_{m}(L, O)}_{m≥1}, after a possible initial string of 1’s, is strictly increasing.

First, for 1 ≤ m ≤ n, we define S_{m}(L, O) ⊂ C_{m}(L, O) as the subset of all simplicial m-configurations (with respect to O). By continuity, C_{m}(L, O) can be replaced by S_{m}(L, O) in the definition of σ_{m}(L, O). We need the following:

Lemma 1. Let {C_{0}, …, C_{m}} ∈ S_{m}(L, O) with O in the relative interior of the m-simplex [C_{0}, …, C_{m}]. Setting

O = ∑_{i=0}^{m} μ_{i}C_{i}, 0 < μ_{i} < 1, i = 0, …, m,

we have

μ_{i} ≤ \frac{1}{1 + Λ(C_{i}, O)}, i = 0, …, m.

For a specific index i, equality holds iff [C_{0}, …, Ĉ_{i}, …, C_{m}] ⊂ ∂L.

Proof. The lemma follows easily by comparing the distortion functions of L and that of the simplex [C_{0}, …, C_{m}] and using (4).

Proof of Proposition 2. Assume that σ_{m−1} = σ_{m}(L, O) for some m ≥ 3. Clearly, O is a regular point. Let {C_{0}, …, C_{m}} ∈ C_{m}(L, O) be minimizing. Thus, [C_{0}, …, C_{m}] is an m-simplex with O in its relative interior, and Lemma 1 applies. Let 0 ≤ i, j ≤ m be distinct, and F_{ij} = ⟨C_{i}, C_{j}, O⟩. Then F_{ij} is an affine plane passing through O. As in Section 3 of [9], superadditivity implies that F_{ij} intersects L in a triangle with vertices C_{i}, C_{j} and another vertex, say, C_{ij}. This latter vertex appears in the decomposition of O in Lemma 1 by compressing all points C_{k}, k ≠ i, j into a single point

μ_{i}C_{i} + μ_{j}C_{j} + μ_{ij}C_{ij} = O with μ_{i} + μ_{j} + μ_{ij} = 1,

where

C_{ij} = \frac{1}{μ_{ij}} ∑_{k=0; k≠i,j}^{m} μ_{k}C_{k}.

Applying Lemma 1 to the triangle [C_{i}, C_{j}, C_{ij}], we obtain

μ_{i} = \frac{1}{1 + Λ(C_{i}, O)} and μ_{j} = \frac{1}{1 + Λ(C_{j}, O)}.

Since i, j are arbitrary, the proposition follows. Finally we will need the following:

Proposition 3. Let C be a smooth point of ∂L and assume that Λ(., O) has a local maximum at C. Then C^{o} is also a smooth point of ∂L and the tangent spaces at C and at C^{o} are parallel. If, in addition, C is a k-flat point (the tangent space at C contains a maximal k-dimensional affine subspace A_{C} and C is contained in the relative interior of ∂L ∩ A_{C}) then C^{o} is ℓ-flat, where ℓ ≥ k, and a translate of A_{C} is contained in A_{C^{o}}.

For the proof, see Section 7 of [9].

3. The measures τ (L, O) and τ_{n−k}(L, O), 1 ≤ k < n

Let L be a compact convex body in a Euclidean vector space E of dimension n, and O an interior point of L. We let S_{m}^{0}(L, O) denote the set of all simplicial m-configurations (with respect to O) with a distinguished element. We usually index the elements of a simplicial m-configuration {B_{0}, …, B_{m}} such that the distinguished element is B_{0}. Then B_{0} is a vertex of the m-simplex [B_{0}, …, B_{m}] with opposite face [B_{1}, …, B_{m}]. We now introduce another sequence of invariants τ_{m}(L, O), 1 < m ≤ n, which will be useful in calculating distortions of cones in the next section. We let

τ_{m}(L, O) = inf_{\{B_{0},...,B_{m}\}∈S_{m}^{0}(L,O)} [\frac{1}{1 + Λ_{[B_{0},...,B_{m}]}(B_{0}, O)} + ∑_{i=1}^{m} \frac{1}{1 + Λ_{L}(B_{i}, O)}] (6)

where Λ_{[B_{0},...,B_{m}]}(., O) is the distortion function of the m-simplex [B_{0}, …, B_{m}] if O is in the relative interior of [B_{0}, …, B_{m}], otherwise it is defined by the obvious limit. In particular, if O ∈ [B_{1}, …, B_{m}] then Λ_{[B_{0},...,B_{n}]}(B_{0}, O) = ∞. Alternatively, Λ_{[B_{0},...,B_{m}]}(B_{0}, O) and Λ_{L}(B_{i}, O), i = 1, …, m, can be interpreted as the distortions of the intersection L ∩ ⟨B_{0}, …, B_{m}⟩ truncated with ⟨B_{1}, …, B_{m}⟩. An easy application of Lemma 1 gives τ_{m}(L, O) ≥ 1. It is also clear from the limiting behavior that

τ_{m}(L, O) ≤ σ_{m−1}(L, O).  (7)

We have τ_{2}(L, O) = 1. For uniformity, we define τ_{1}(L, O) = 1. (It seems to be a difficult problem to decide whether equality holds in (7).) By (3) and (7), we have

1 ≤ τ_{m}(L, O) ≤ \frac{m}{2}.

τ_{m}(L, O) = 1 if and only if σ_{m−1}(L, O) = 1. (Once again, see Lemma 1.) Also, if τ_{m}(L, O) = m/2 for some m ≥ 3, then L is symmetric with respect to O. As usual, we suppress the index m = n. With this, we have

τ_{m}(L, O) = inf_{O∈F⊂E ; dim F=m} τ(L ∩ F, O),  (8)

where the infimum is over affine subspaces F ⊂ E. Clearly, (5) holds for σ replaced by τ:

τ_{m′}(L, O) ≤ τ_{m}(L, O) + \frac{m′ − m}{1 + max_{∂L} Λ(., O)}, 1 ≤ m ≤ m′ ≤ n.  (9)

It will be convenient to define the out-of-range invariant τ_{n+1}(L, O) = σ(L, O).

Given {B_{0}, …, B_{m}} ∈ S_{m}^{0}(L, O), we define the bulging

β_{[B_{0},...,B_{m}]}(B_{i}, O) = \frac{1}{1 + Λ_{L}(B_{i}, O)} − \frac{1}{1 + Λ_{[B_{0},...,B_{n}]}(B_{i}, O)}.  (10)

We then have

τ_{m}(L, O) = inf_{{B_{0},...,B_{m}}∈S_{m}^{0}(L,O)} ∑_{i=1}^{m} β_{[B_{0},...,B_{m}]}(B_{i}, O) + 1.  (11)

Proposition 4. For 1 < m ≤ n, we have

mτ_{m−1}(L, O) ≤ (m − 1)τ_{m}(L, O) + 1.  (12)

Proof. Let ϵ > 0 and consider a simplicial m-configuration {B_{0}, …, B_{m}} with respect to O such that

\frac{1}{1 + Λ_{[B_{0},...,B_{m}]}(B_{0}, O)} + ∑_{i=1}^{m} \frac{1}{1 + Λ_{L}(B_{i}, O)} < τ_{m}(L, O) + ϵ.  (13)

We may assume that O is in the relative interior of the m-simplex [B_{0}, …, B_{m}]. With this, we have

O = ∑_{i=0}^{m} λ_{i}B_{i}, ∑_{i=0}^{m} λ_{i} = 1, 0 < λ_{i} < 1.

We write

O = (λ_{0} + λ_{1})B_{01} + ∑_{i=2}^{m} λ_{i}B_{i},

where

B_{01} = \frac{λ_{0}}{λ_{0} + λ_{1}} B_{0} + \frac{λ_{1}}{λ_{0} + λ_{1}} B_{1}.

Let B̄_{01} = μB_{01} ∈ ∂L, where μ ≥ 1. Clearly, [B̄_{01}, B_{2}, …, B_{m}] is an (m − 1)-simplex with O in its interior. Let B̄_{01}^{o} ∈ [B_{2}, …, B_{m}] be the opposite of B̄_{01} with respect to this simplex. [B_{0}, B_{1}, B̄_{01}^{o}] is a triangle with O in its interior. We now add the terms

\frac{1}{1 + Λ_{[B̄_{01},B_{2},...,B_{m}]}(B̄_{01}, O)} + \frac{1}{1 + Λ_{[B̄_{01},B_{2},...,B_{m}]}(B̄_{01}^{o}, O)} − 1

(amounting to zero) to the left-hand side of the inequality in (13), and split the terms into two groups. The terms

\frac{1}{1 + Λ_{[B̄_{01},B_{2},...,B_{m}]}(B̄_{01}, O)} + ∑_{i=2}^{m} \frac{1}{1 + Λ_{L}(B_{i}, O)} − 1

can be estimated below by τ_{m−1}(L, O) − 1 since B̄_{01} can be taken as the distinguished element in the (m − 1)-configuration {B̄_{01}, B_{2}, …, B_{m}}. The remaining terms in the second group are

\frac{1}{1 + Λ_{[B_{0},...,B_{m}]}(B_{0}, O)} + \frac{1}{1 + Λ_{L}(B_{1}, O)} + \frac{1}{1 + Λ_{[B̄_{01},B_{2},...,B_{m}]}(B̄_{01}^{o}, O)}. (14)

The first term in (14) is equal to

\frac{1}{1 + Λ_{[B̄_{01}^{o},B_{0},B_{1}]}(B_{0}, O)}.

By the same token, the second term in (14) is equal to

β_{[B_{0},...,B_{m}]}(B_{1}, O) + \frac{1}{1 + Λ_{[B̄_{01}^{o},B_{0},B_{1}]}(B_{1}, O)}.

Finally, the third term in (14) can be estimated as

\frac{1}{1 + Λ_{[B̄_{01},B_{2},...,B_{m}]}(B̄_{01}^{o}, O)} ≥ \frac{1}{1 + Λ_{[B̄_{01}^{o},B_{0},B_{1}]}(B̄_{01}^{o}, O)}

since

Λ_{[B̄_{01},B_{2},...,B_{m}]}(B̄_{01}^{o}, O) ≤ Λ_{[B̄_{01}^{o},B_{0},B_{1}]}(B̄_{01}^{o}, O).

Putting everything together, we obtain

τ_{m−1}(L, O) + β_{[B_{0},...,B_{m}]}(B_{1}, O) ≤ τ_{m}(L, O) + ϵ

since the terms involving distortions at the vertices of the triangle [B̄_{01}^{o}, B_{0}, B_{1}] add up to 1. Replacing B_{1} by B_{i}, summing up with respect to i = 1, …, m, and letting ϵ → 0, we obtain

mτ_{m−1}(L, O) + ∑_{i=1}^{m} β_{[B_{0},...,B_{m}]}(B_{i}, O) ≤ mτ_{m}(L, O).

Finally, using (11), we arrive at (12). Since τ_{m}(L, O) ≥ 1, (12) implies that the sequence {τ_{m}(L, O)}_{m=1}^{n} is nondecreasing. Moreover, again by (12), the only way τ_{m−1}(L, O) = τ_{m}(L, O) can happen is that it is equal to 1. This means that the sequence {τ_{m}(L, O)}_{m=1}^{n}, after an initial string of 1’s, is strictly increasing. Iterating (12), for 1 ≤ m′ < m ≤ n, we get

mτ_{m−m′}(L, O) ≤ (m − m′)τ_{m}(L, O) + m′.

Replacing m′ by m − m′ and adding, we obtain

τ_{m−m′}(L, O) + τ_{m′}(L, O) ≤ τ_{m}(L, O) + 1,

a subadditive property.

4. Convex cones

Let L_{0} ⊂ E_{0} be a compact convex body and O_{0} an interior point of L_{0}. Let E = E_{0} × R and O_{1} ∈ E not contained in E_{0}. We consider the cone L = [L_{0}, O_{1}]. We let 0 < λ < 1 and O_{λ} = (1 − λ)O_{0} + λO_{1}. In this section we calculate σ(L, O_{λ}) in terms of L_{0}. We assume that L_{0} and therefore L are not simplicial since otherwise there is nothing to calculate. It is technically convenient to assume that O_{0} is the origin of E_{0}. We also set n = dim L so that dim L_{0} = n − 1.

Theorem D. Let L be a cone with base L_{0} and vertex O_{1}. Let O_{λ} = (1 − λ)O_{0} + λO_{1}, 0 < λ < 1, be the base point of L, where O_{0} is the base point of L_{0}. Then we have the following: I. If 0 < λ ≤ \frac{1}{2+max_{∂L_{0}} Λ(.,O_{0})} then

σ(L, O_{λ}) = min_{1≤m≤n} ((n − m + 1)λ + (1 − λ)τ_{m}(L_{0}, O_{0})) ;

II. If \frac{1}{2+max_{∂L_{0}} Λ(.,O_{0})} ≤ λ < 1 then

σ(L, O_{λ}) = λ + (1 − λ) min_{1≤m≤n} (τ_{m}(L_{0}, O_{0}) + \frac{n − m}{1 + max_{∂L_{0}} Λ(., O_{0})})
= λ + (1 − λ) min (σ(L_{0}, O_{0}), τ(L_{0}, O_{0}) + \frac{1}{1 + max_{∂L_{0}} Λ(., O_{0})}) .

In these formulas, τ_{n}(L_{0}, O_{0}) = σ_{n−1}(L_{0}, O_{0}) = σ(L_{0}, O_{0}). In particular, the function λ ↦ σ(L, O_{λ}), λ ∈ [0, 1], is piecewise linear and concave (with limit equal to 1 at the endpoints).

The proof of Theorem D is technical. In what follows we will make a number of observations and reduce Theorem D to a simpler one. To simplify the notation we let γ denote the function λ ↦ σ(L, O_{λ}), λ ∈ [0, 1]. In addition, for 1 ≤ m ≤ n, we consider the linear functions

α_{m}(λ) = (n − m + 1)λ + (1 − λ)τ_{m},
β_{m}(λ) = λ + (1 − λ) (τ_{m} + \frac{n − m}{1 + M_{0}} ) ,

where we suppressed L_{0}, O_{0}, and set M_{0} = max_{∂L_{0}} Λ(., O_{0}). Recall here also that τ_{1} = τ_{2} = 1, τ_{n−1} = τ, and τ_{n} = σ_{n−1} = σ. With these, I–II of Theorem D rewrites as I. If 0 < λ ≤ 1/(2 + M_{0}) then γ(λ) = min_{1≤m≤n} α_{m}(λ). II. If 1/(2 + M_{0}) ≤ λ < 1 then γ(λ) = min_{1≤m≤n} β_{m}(λ). Clearly, α_{n} = β_{n}. Moreover, for 1 ≤ m < n, we have

α_{m}(0) = τ_{m} < τ_{m} + \frac{n − m}{1 + M_{0}} = β_{m}(0) and α_{m}(1) = n − m + 1 > 1 = β_{m}(1),

while α_{m} and β_{m} attain the same value at 1/(2 + M_{0}). It follows that in both cases I–II, we have

γ(λ) = min_{1≤m≤n} (α_{m}(λ), β_{m}(λ)) , λ ∈ [0, 1].

Thus, γ is piecewise linear and concave. By (2), the one-sided limits of γ at the endpoints of [0, 1] are both equal to 1. The second statement of Theorem D follows. By (9), for 1 ≤ m < m′ < n, we have β_{m} ≥ β_{m′}. Thus,

min_{1≤m≤n} β_{m}(λ) = min(β_{n}(λ), β_{n−1}(λ)).

This is the second equality in case B of Theorem D. We now give a more detailed analysis of the minimum in case I:

γ(λ) = min_{1≤m≤n} α_{m}(λ), λ ∈ [0, 1/(2 + M_{0})].

For 1 ≤ m < m′ ≤ n, the linear functions α_{m} and α_{m′} have the same value at

ℓ_{m,m′} = 1 − \frac{1}{1 + \frac{τ_{m′}−τ_{m}}{m′−m}} .  (15)

For 1 ≤ m < m′ < n, using (9), we have

ℓ_{m,m′} ≤ 1 − \frac{1}{1 + \frac{1}{1+M_{0}}} = \frac{1}{2 + M_{0}} .

We say that α_{m} participates in γ if the zero set of (α_{m}−γ)|_{[0,1/(2+M_{0})]} has nonempty interior. Since γ is concave, this zero set is a closed interval. We call this the interval of participation of α_{m} in γ. Since γ is piecewise linear, the interval [0, 1/(2 + M_{0})] is subdivided into intervals of participation for the various α_{m}, 1 ≤ m ≤ n. We have

1 = α_{1}(0) = τ_{1} = α_{2}(0) = τ_{2} ≤ ⋯ ≤ α_{n−1}(0) = τ ≤ σ_{n−2} ≤ σ = τ_{n} = α_{n}(0),

and

α′_{1}(0) = n − 1 > α′_{2}(0) = n − 2 > ⋯ > α′_{n−1}(0) = 2 − τ ≥ α′_{n}(0) = 1 − σ.

Let 1 ≤ m < m′ ≤ n, and assume that both α_{m} and α_{m′} participate in γ. Comparing the intercepts and slopes above, we see that concavity of γ implies that the interval of participation of α_{m} precedes that of α_{m′}. We let 1 < m_{1} < ⋯ < m_{s} ≤ n denote those indices m for which α_{m} participates in γ. By the above, the intervals of participation of α_{m_{i}}, i = 1, …, s, subdivide [0, 1/(2 + M_{0})] in a successive manner. Comparing the slopes and intercepts above we see that m_{1} is the largest index m such that τ_{m} = 1. (Recall that the sequence {τ_{m}}_{m=1}^{n} starts with a string of 1’s.)

For 1 ≤ i < s, the interval of participation of α_{m_{i}} in γ is [ℓ_{m_{i−1},m_{i}}, ℓ_{m_{i},m_{i+1}}]. (Here we set m_{0} = 0 so that ℓ_{m_{0},m_{1}} = 0.) The last interval of participation is therefore [ℓ_{m_{s−1},m_{s}}, 1/(2 + M_{0})]. By continuity of γ, at λ = 1/(2 + M_{0}), the matching condition

α_{m_{s}}(λ) = min(β_{n}(λ), β_{n−1}(λ))

must be satisfied. This works out to be

min (σ, τ + \frac{1}{1 + M_{0}} ) = τ_{m_{s}} + \frac{n − m_{s}}{1 + M_{0}} .

If σ < τ + 1/(1 + M_{0}) then m_{s} = n. If σ ≥ τ + 1/(1 + M_{0}) then, for m_{s} = n, we have σ = τ + 1/(1 + M_{0}), and, for m_{s} < n, we have τ = τ_{m_{s}} + (n − 1 − m_{s})/(1 + M_{0}). There are several consequences of this discussion.

Theorem E. Assume that τ_{3}(L_{0}, .), …, τ_{n−1}(L_{0}, .) = τ(L_{0}, .), σ(L_{0}, .) are all concave on int L_{0}. Then σ(L, .) is also concave on int L.

Proof. Since the minimum of concave functions is concave, we need to show that α_{m} and β_{m}, m = 1, …, n, are concave functions on int L. We do this in a slightly more general setting. Assume that ϕ is a concave function on int L_{0}, and c > 0 is a fixed constant. Define the function ψ : int L → R by

ψ((1 − λ)O_{0} + λO_{1}) = cλ + (1 − λ)ϕ(O_{0}), O_{0} ∈ int L_{0}, 0 < λ < 1.

Since α_{m} and β_{m} are special cases of this (see also the corollary to Proposition 1 in [9]), it remains to show that ψ is concave. We let O_{0}^{0}, O_{0}^{1} ∈ int L_{0}, 0 < μ, ν < 1, and

O_{μ}^{0} = (1 − μ)O_{0}^{0} + μO_{1} and O_{ν}^{1} = (1 − ν)O_{0}^{1} + νO_{1}.

We need to show that

ψ((1 − κ)O_{μ}^{0} + κO_{ν}^{1}) ≥ (1 − κ)ψ(O_{μ}^{0}) + κψ(O_{ν}^{1}), 0 < κ < 1.

We write

(1 − κ)O_{μ}^{0} + κO_{ν}^{1} = (1 − λ) (\frac{(1 − κ)(1 − μ)}{1 − λ}O_{0}^{0} + \frac{κ(1 − ν)}{1 − λ}O_{0}^{1}) + λO_{1},

where λ = (1 − κ)μ + κν. With these, we have

ψ((1 − λ)O_{0} + λO_{1}) = cλ + (1 − λ)ϕ (\frac{(1 − κ)(1 − μ)}{1 − λ}O_{0}^{0} + \frac{κ(1 − ν)}{1 − λ}O_{0}^{1})
≥ cλ + (1 − κ)(1 − μ)ϕ(O_{0}^{0}) + κ(1 − ν)ϕ(O_{0}^{1})
= (1 − κ)(cμ + (1 − μ)ϕ(O_{0}^{0})) + κ(cν + (1 − ν)ϕ(O_{0}^{1}))
= (1 − κ)ψ(O_{μ}^{0}) + κψ(O_{ν}^{1}).

Concavity and Theorem E follow.

Note that Theorem E (along with Theorem A) implies the first statement of Theorem B. For the second statement we have the following:

Example. Let Δ ⊂ R^{2} be an equilateral triangle inscribed in the unit circle of R^{2}. Let L_{0} be the intersection of the vertical cylinder in R^{3} with base Δ and the unit ball. Then σ_{2}(L_{0}, .) is not concave. Indeed, for O ∈ Δ, we have σ_{2}(L_{0}, O) = 1, and, for any other point O in the interior of L_{0}, we have σ_{2}(L_{0}, O) > 1 as there is no triangular intersection of L_{0} away from Δ. Now consider L_{0} as the base of a 4-dimensional cone L (with vertex O_{1}). We claim that σ(L, .) is not concave. Let O_{t} = (0, 0, t), |t| < 1. We calculate σ(L, (1−λ)O_{t} +λO_{1}) for small λ > 0. For t = 0, O_{0} ∈ Δ. Since Δ is a triangular intersection of L_{0}, we have σ_{2}(L_{0}, O_{0}) = 1. Thus, we have

1 = τ_{1}(L_{0}, O_{0}) = τ_{2}(L_{0}, O_{0}) = τ(L_{0}, O_{0}) < τ_{4}(L_{0}, O_{0}) = σ(L_{0}, O_{0}).

Hence, the last 1 in this sequence is at the index m_{1} = 3. Consequently, for small λ > 0, we have

σ(L, (1 − λ)O_{0} + λO_{1}) = α_{3}(λ) = 2λ + (1 − λ)τ(L_{0}, O_{0}) = 1 + λ.

Now let t ≠ 0. Since O_{t} ∉ Δ, L_{0} has no triangular intersection passing through O_{t}, we have σ_{2}(L_{0}, O_{t}) > 1. Thus

1 = τ_{1}(L_{0}, O_{t}) = τ_{2}(L_{0}, O_{t}) < τ(L_{0}, O_{t}) ≤ τ_{4}(L_{0}, O_{t}) = σ(L_{0}, O_{t}).

The last 1 in this sequence has index m_{1} = 2. Consequently, for small λ > 0, we have

σ(L, (1 − λ)O_{t} + λO_{1}) = α_{2}(λ) = 3λ + (1 − λ)τ_{2}(L_{0}, O_{0}) = 1 + 2λ.

(Note that the length of the first interval of participation tends to zero as t → 0.) Let 0 < t < 1 be fixed and λ > 0 small enough so that the formulas above hold for ±t. Then, at the endpoints of the line segment [O_{−t}, O_{t}], the function σ(L, .) is 1 + 2λ and at the midpoint O_{0} it is 1 + λ. Thus, σ(L, .) is not concave. It remains to prove Theorem D. To do this, in view of the discussion above, we need to show the following:

Theorem F. Let L, L_{0} and O_{0} be as in Theorem D. We have the following: (i) If O_{λ} ∈ R is a regular point then

σ(L, O_{λ}) = λ + (1 − λ)σ(L_{0}, O_{0}).  (16)

(ii) If O_{λ} ∈ S is singular with degree of singularity n − m then

σ(L, O_{λ}) = λ + (1 − λ)τ_{m}(L_{0}, O_{0}) + \frac{n − m}{1 + max_{∂L} Λ(., O_{λ})},  (17)

where

\frac{1}{1 + max_{∂L} Λ(., O_{λ})} = \{ \begin{matrix}λ, & if 0 < λ ≤ \frac{1}{2+max_{∂L_{0}} Λ(.,O_{0})} \\ \frac{1−λ}{1+max_{∂L_{0}} Λ(.,O_{0})}, & if \frac{1}{2+max_{∂L_{0}} Λ(.,O_{0})} ≤ λ < 1 \end{matrix} .

(18)

In addition, the infimum in τ_{m}(L_{0}, O_{0}) is attained by a simplicial configuration.

The proof of Theorem F is given in the rest of this section. We begin with the following:

Lemma 2. Let A_{0}, A_{1} ∈ R^{2} be distinct points and A_{λ} = (1 − λ)A_{0} + λA_{1} with 0 < λ < 1. Let B_{0}, B_{λ}, B_{1} be collinear points perspectively related to A_{0}, A_{λ}, A_{1} by a perspectivity with center at O. Let

Λ_{0} = \frac{d(A_{0}, O)}{d(B_{0}, O)}, Λ_{1} = \frac{d(A_{1}, O)}{d(B_{1}, O)}, and Λ_{λ} = \frac{d(A_{λ}, O)}{d(B_{λ}, O)}.

Then, we have

Λ_{λ} = (1 − λ)Λ_{0} + λΛ_{1}.

Proof. We may assume that O is the origin. Then, we have

B_{0} = ±\frac{1}{Λ_{0}} A_{0}, B_{1} = ±\frac{1}{Λ_{1}} A_{1}, and B_{λ} = ±\frac{1}{Λ_{λ}} A_{λ}.

We also have B_{λ} = (1 − μ)B_{0} + μB_{1} for some 0 < μ < 1. Playing this equation back to the definition of A_{λ} and comparing, Lemma 2 follows.

Corollary. Let [B_{0}, B_{1}, C] ⊂ R^{2} be a triangle and choose points O_{0} and O_{1} in the interior of the sides [B_{0}, C] and [B_{1}, C]. Let 0 < λ < 1 and O_{λ} = (1−λ)O_{0}+λO_{1}. Consider the distortions

Λ_{0} = \frac{d(C, O_{0})}{d(B_{0}, O_{0})}, Λ_{1} = \frac{d(C, O_{1})}{d(B_{1}, O_{1})}, Λ_{λ} = \frac{d(C, O_{λ})}{d(B_{λ}, O_{λ})},

where B_{λ} is the projection of O_{λ} to the side [B_{0}, B_{1}] from C. Then we have

\frac{1}{1 + Λ_{λ}} = \frac{1 − λ}{1 + Λ_{0}} + \frac{λ}{1 + Λ_{1}} .

Proof. O_{0}, O_{λ}, O_{1} and B_{0}, B_{λ}, B_{1} are perspectively related with perspectivity centered at C. The corollary now follows from Lemma 2.

Proposition 5. Let C_{0} ∈ ∂L_{0}, 0 ≤ t ≤ 1, and define C = (1 − t)C_{0} + tO_{1}. Then we have

\frac{1}{1 + Λ(C, O_{λ})} = \{ \begin{matrix}\frac{1−λ}{1−t} \frac{1}{1+Λ(C_{0},O_{0})}, & if 0 ≤ t ≤ t^{∗} \\ \frac{λ}{t}, & if t^{∗} ≤ t ≤ 1 \end{matrix}

where

t^{∗} = 1 − \frac{1 − λ}{1 + λΛ(C_{0}, O_{0})}  (19)

is defined by the condition that, for t = t^{∗}, the opposite of C with respect to O_{λ} in on ∂L_{0}.

Let C be as in the proposition. We call C a type I point if 0 < t < t^{∗}, a type II point if t^{∗} < t < 1 and a type III point if t = t^{∗}. In addition, we call a point C ∈ ∂L_{0} type 0. This corresponds to the parameter value t = 0.

Proof of Proposition 5. Let C^{o} be the opposite of C with respect to O_{λ} in L, and C_{0}^{o} the opposite of C_{0} with respect to O_{0} in L_{0}. The critical value t^{∗} is defined by C^{o} = C_{0}^{o}. We first let 0 ≤ t ≤ t^{∗}. Since C^{o} ∈ [C_{0}^{o}, O_{1}], we have C^{o} = (1 − s)C_{0}^{o} + sO_{1}, for some 0 ≤ s ≤ 1. Since C, C^{o}, O_{λ} are collinear, we have

O_{λ} = (1 − μ)C + μC^{o},

where 0 < μ < 1. Expanding, using C_{0}^{o} = −C_{0}/Λ(C_{0}, O_{0}), and comparing coefficients, we have

(1 − μ)t + μs = λ
(1 − μ)(1 − t) − μ\frac{1 − s}{Λ(C_{0}, O_{0})} = 0.

Eliminating s we obtain

μ = \frac{(1 − t)Λ(C_{0}, O_{0}) + λ − t}{(1 − t)(1 + Λ(C_{0}, O_{0}))}.

With this, we have

Λ(C, O_{λ}) = \frac{μ}{1 − μ} = \frac{1 − t}{1 − λ} (Λ(C_{0}, O_{0}) + 1) − 1.

The proposition follows in this case. The formula for the critical value t^{∗} also follows since this corresponds to s = 0. Now let t^{∗} ≤ t ≤ 1 and C^{∗} = (1−t^{∗})C_{0}+t^{∗}O_{1} the corresponding point. The pencil of points C^{∗}, C, O_{1} and C_{0}^{o}, C^{o}, O_{0} are perspectively related by the perspectivity with center at O_{λ}. By Lemma 2, we have

Λ(C, O_{λ}) = (1 − ν)Λ(C^{∗}, O_{λ}) + νΛ(O_{1}, O_{λ}),

where

C = (1 − ν)C^{∗} + νO_{1}.

Comparing this last equation with C = (1−t)C_{0}+tO_{1} and C^{∗} = (1−t^{∗})C_{0}+t^{∗}O_{1}, and using the value of t^{∗}, we obtain

ν = 1 − \frac{1 − t}{1 − λ}(1 + Λ(C_{0}, O_{0})).

Substituting this into the formula of Λ(C, O_{λ}) above, the proposition follows.

We now return to the cone L with base L_{0}. Let C ∈ ∂L_{0}. Let C_{0} and C_{λ} be the opposites of C with respect to O_{0} and O_{λ} in L_{0} and L. In Corollary to Lemma 2 we set B_{0} = C_{0}, B_{λ} = C_{λ} and B_{1} = O_{1} so that Λ_{1} = ∞. We thus obtain

\frac{1}{1 + Λ_{L}(C, O_{λ})} = \frac{1 − λ}{1 + Λ_{L_{0}}(C, O_{0})}, C ∈ ∂L_{0}.  (20)

Note that this proves (18). Indeed, by Proposition 3, the local maxima of Λ(., O_{λ}) are located at O_{1} or along ∂L_{0}. Thus, (18) follows from (20) and from the obvious fact that λ = 1/(1 + Λ(O_{1}, O_{λ})). We now split the proof proof according to whether O_{λ} is regular or singular.

Proposition 6. Let L be a cone with base L_{0} as above. Assume that O_{λ} ∈ R. Then O_{0} ∈ R_{0}, where R_{0} is the regular set of L_{0}, and

σ(L, O_{λ}) = λ + (1 − λ)σ(L_{0}, O_{0}).  (21)

Proof. Let {C_{0}, …, C_{n}} ∈ C(L, O_{λ}) be minimal. By Proposition 1, [C_{0}, …, C_{n}] is an n-simplex with O_{λ} in its interior and, at each C_{i}, Λ(., O_{λ}) attains its local maximum. By Proposition 3, the possible local maxima are located at the vertex O_{1} of the cone L_{0} or along the boundary of the base L_{0}. Thus, one of the points in the configuration must be O_{1}.Without loss of generality, we may assume that C_{0} = O_{1}. It also follows that C_{1}, …, C_{n} ∈ ∂L_{0}. Since O_{λ} is in the interior of the n-simplex [C_{0}, …, C_{n}], the point O_{0} is in the interior of the (n − 1)-simplex [C_{1}, …, C_{n}]. We claim that the configuration {C_{1}, …, C_{n}} ∈ C(L_{0}, O_{0}) is minimal. Otherwise, let {C′_{1}, …, C′_{n}} ∈ C(L_{0}, O_{0}) be a minimal configuration so that

∑_{i=1}^{n} \frac{1}{1 + Λ(C′_{i}, O_{0})} < ∑_{i=1}^{n} \frac{1}{1 + Λ(C_{i}, O_{0})}.

By (20), we also have

∑_{i=1}^{n} \frac{1}{1 + Λ(C′_{i}, O_{λ})} < ∑_{i=1}^{n} \frac{1}{1 + Λ(C_{i}, O_{λ})}.

Adding \frac{1}{1+Λ(C_{0},O_{λ})} to both sides, we obtain a contradiction to the minimality of {C_{0}, …, C_{n}}. The claim follows.

Using (20), we have

σ(L, O_{λ}) = ∑_{i=0}^{n} \frac{1}{1 + Λ(C_{i}, O_{λ})}

= \frac{1}{1 + Λ(C_{0}, O_{λ})} + ∑_{i=1}^{n} \frac{1}{1 + Λ(C_{i}, O_{λ})}
= λ + (1 − λ)σ(L_{0}, O_{0}).

Finally, working backwards, it is clear that every minimal configuration {C_{1}, …, C_{n}} ∈ C(L_{0}, O_{0}) is an (n − 1)-simplex with O_{0} in its interior as its extension by adjoining C_{0} = O_{1} is minimal. Thus, we have O_{0} ∈ R_{0}. The proposition follows.

In the remainder of this section we study the case when O_{λ} ∈ S is a singular point in L. Assume that the degree of singularity of O_{λ} is n − m, m < n. By Proposition 1, there is a minimal n-configuration {C_{0}, …, C_{n}} ∈ C(L, O_{λ}) which contains a simplicial minimal m-configuration {C_{0}, …, C_{m}} ∈ C_{m}(L, O_{λ}), m is the least number with this property, the point O_{λ} is in the relative interior of the msimplex [C_{0}, …, C_{m}], and Λ(., O_{λ}) attains its absolute maximum at C_{m+1}, …, C_{n} in ∂L. We can exclude the trivial case m = 1 since then the configuration contains n points at which Λ(., O_{λ}) attains its maximum and an additional antipodal point. Thus, from now on we assume that m ≥ 2. We have

σ(L, O_{λ}) = σ_{m}(L, O_{λ}) + \frac{n − m}{1 + max_{∂L} Λ(., O_{λ})},  (22)

where

σ_{m}(L, O_{λ}) = ∑_{i=0}^{m} \frac{1}{1 + Λ(C_{i}, O_{λ})}.

Let F = ⟨C_{0}, …, C_{m}⟩; it is an affine subspace of dimension m. F is contained in the affine subspace ⟨F, O_{1}⟩, properly (with codimension 1) iff O_{1} ∉ F. The intersection L ∩ ⟨F, O_{1}⟩ is a cone with base L_{0} ∩ ⟨F, O_{1}⟩ ∋ O_{0} and vertex O_{1}. It is clear that {C_{0}, …, C_{m}} is not only minimal in L but also in L ∩ ⟨F, O_{1}⟩, in particular

σ_{m}(L ∩ ⟨F, O_{1}⟩, O_{λ}) = σ_{m}(L, O_{λ}).

Case I. O_{1} ∈ F. We claim that O_{λ} is a regular point in L ∩ ⟨F, O_{1}⟩. Assume, on the contrary, that there is a minimal m-configuration {C′_{0}, …, C′_{m}} ∈ C_{m}(L ∩ ⟨F, O_{1}⟩, O_{λ}) which contains a k-configuration with k < m. Since

∑_{i=0}^{m} \frac{1}{1 + Λ(C′_{i}, O_{λ})} = σ_{m}(L ∩ ⟨F, O_{1}⟩, O_{λ}) = σ_{m}(L, O_{λ})

the extended configuration {C′_{0}, …, C′_{m}, C_{m+1}, …, C_{n}} ∈ C_{n}(L, O_{λ}) is minimal. By assumption, it contains a k-configuration. Thus, O_{λ} ∈ S_{n−k} so that k ≥ m. This is a contradiction.

We now apply Proposition 6 to the regular point O_{λ} of L ∩ ⟨F, O_{1}⟩. We obtain

σ_{m}(L, O_{λ}) = λ + (1 − λ)σ_{m−1}(L_{0}, O_{0}),  (23)

where σ(L_{0} ∩ ⟨F, O_{1}⟩, O_{0}) = σ_{m−1}(L_{0}, O_{0}). Combining (22) and (23), we obtain

σ(L, O_{λ}) = λ + (1 − λ)σ_{m−1}(L_{0}, O_{0}) + \frac{n − m}{1 + max_{∂L} Λ(., O_{λ})}.

By (7) this does not compete in the infimum in (17) unless equality holds in (7). Case II. O_{1} ∉ F. We now study what type of points are possible in the simplicial minimal configuration {C_{0}, …, C_{m}}. Let C_{i} and C_{j} be two distinct configuration points and assume that

C_{i} = (1 − t_{i})C_{i,0} + t_{i}O_{1} and C_{j} = (1 − t_{j})C_{j,0} + t_{j}O_{1},

where C_{i,0}, C_{j,0} ∈ ∂L_{0} and 0 < t_{i}, t_{j} < 1. By definition, C_{i}, C_{j} can be type I, type II, or type III points. We define a variation in which C_{i} and C_{j} move simultaneously along the line segments [C_{i,0}, O_{1}] and [C_{j,0}, O_{1}] while keeping the condition O_{λ} ∈ [C_{0}, …, C_{m}] intact. We first write

∑_{k=0}^{m} μ_{k}C_{k} = O_{λ},

where ∑_{k=0}^{m} μ_{k} = 1 with 0 < μ_{k} < 1, k = 0, …, m. For s small, we define

C_{i}(s) = \frac{1}{1 + s}C_{i} + \frac{s}{1 + s}O_{1}
C_{j}(s) = \frac{μ_{j}}{μ_{j} − μ_{i}s}C_{j} − \frac{μ_{i}s}{μ_{j} − μ_{i}s}O_{1}.

Setting

μ_{i}(s) = μ_{i}(1 + s) and μ_{j}(s) = μ_{j} − μ_{i}s,

we have

μ_{i}(s)C_{i}(s) + μ_{j}(s)C_{j}(s) + ∑_{k=0, k≠i,j}^{m} μ_{k}C_{k} = O_{λ}.

Thus, substituting C_{i}(s) and C_{j}(s) for C_{i} and C_{j}, the condition O_{λ} ∈ [C_{0}, …, C_{m}] remains in effect. The parameter values change as

t_{i}(s) = \frac{t_{i} + s}{1 + s} and t_{j}(s) = \frac{μ_{j}t_{j} − μ_{i}s}{μ_{j} − μ_{i}s}.

Finally, we need to see how

\frac{1}{1 + Λ(C_{i}, O_{λ})} + \frac{1}{1 + Λ(C_{j}, O_{λ})}

changes under this substitution.

First, let C_{i} be type I and C_{j} type II or type III. We set s < 0 so that C_{i} stays type I and C_{j} becomes or stays type III. We have

\frac{1}{1 + Λ(C_{i}(s), O_{λ})} + \frac{1}{1 + Λ(C_{j}(s), O_{λ})}

\frac{1 − λ}{1 − t_{i}(s)} \frac{1}{1 + Λ(C_{i}(s), O_{λ})} + \frac{λ}{t_{j}(s)}

= \frac{1}{1 + Λ(C_{i}, O_{λ})} + \frac{1}{1 + Λ(C_{j}, O_{λ})}

+ s\frac{1 − λ}{1 − t_{i}} \frac{1}{1 + Λ(C_{i,0}, O_{0})}

+ \frac{λ}{1 − μ_{j} \frac{1−t_{j}}{μ_{j}−μ_{i}s}} − \frac{λ}{t_{j}} .

For s < 0 the last three terms are negative. This contradicts minimality. We conclude that if there is a type II or III point then there cannot be a type I point. Assume now that there are no type II and type III points. It is clear that there is at least one type I point since otherwise the condition O_{λ} ∈ [C_{0}, …, C_{m}] could not be satisfied. We claim that actually there are at least two type I points. Suppose that there is only one type I point C_{i}. Then the line ⟨C_{i}, O_{λ}⟩ intersects the i-th face of the simplex [C_{0}, …, C_{m}] opposite to C_{i}, and this face is entirely contained in L_{0}. Based on the geometric meaning of type II and type III points in Proposition 5, C_{i} cannot be type I. Now let C_{i} and C_{j} be both type I. Performing the variation as above, a brief computation shows that

\frac{1}{1 + Λ(C_{i}(s), O_{λ})} + \frac{1}{1 + Λ(C_{j}(s), O_{λ})}

= \frac{1}{1 + Λ(C_{i}, O_{λ})} + \frac{1}{1 + Λ(C_{j}, O_{λ})}

+s (\frac{1 − λ}{1 − t_{i}} \frac{1}{1 + Λ(C_{i,0}, O_{0})} − \frac{μ_{i}}{μ_{j}} \frac{1 − λ}{1 − t_{j}} \frac{1}{1 + Λ(C_{j,0}, O_{0})}) .

The condition of minimality implies that the last term in the parentheses must be zero. This means that the sum

\frac{1}{1 + Λ(C_{i}(s), O_{λ})} + \frac{1}{1 + Λ(C_{j}(s), O_{λ})}

stays constant for all admissible s. Assume, for definiteness, that s > 0 increases. This means that C_{i}(s) moves away from C_{i,0} while C_{j}(s) approaches C_{j,0}. The point C_{i}(s) cannot hit the type III territory before C_{j,0} hits ∂L_{0} since otherwise we would apply the previous variation and get a contradiction. Thus, at a critical value of s, C_{j}(s) ∈ ∂L_{0}. Applying this to every type I point we will be left with only one type I point which again gives a contradiction.

Summarizing, we obtained that there cannot be any type I points among {C_{0}, …, C_{m}}. It also follows that there must be at least one type II or type III point. We now claim that there are no configuration points in the relative interior of L_{0} ∩ F. Suppose that C_{i} is in the relative interior of L_{0} ∩ F. Then, by definition, C_{i} is an (m − 1)-flat point in L ∩ ⟨F, O_{1}⟩. By Proposition 3, the antipodal must be (at least an) m-flat (since it is on the boundary of a cone away from the base and the vertex), and the (m − 1)-flat translates into the m-flat. We can thus move C_{i} within this flat to any point without changing the distortion. Note also that during the move the condition O_{λ} ∈ [C_{0}, …, C_{m}] stays intact due to the minimality of m. On the other hand, the antipodal of a type II or type III point is in this flat. Thus, we can move C_{i} to this antipodal and obtain an antipodal pair of points in the configuration. This means that m = 1, and this trivial case was excluded. We now claim that there is no type II point and that there is a unique type III point. Let C = C_{i} be any type II or type III point with C = (1 − t)C_{0} + tO_{1}, C_{0} ∈ ∂L_{0} and 0 < t < 1. (Here we suppress the index i and briefly revert to the notation of Proposition 5; in particular, C_{0} is the point corresponding to C in ∂L_{0} not the first point in the configuration.) We assume that Λ(., O_{λ}) restricted to F attains a local maximum at C. In what follows, we will work within the subspace ⟨F, O_{1}⟩ in which F is a codimension one affine subspace. Let N be the unit normal vector to the hyperplane F pointing to the half-space that contains O_{1}. We first claim that the function on ∂L_{0} defined by C′ ↦ ⟨C′, N⟩ attains a local minimum at C_{0}. Central projection from O_{1} gives a one-to-one correspondence between a neighborhood of C in ∂L∩F and a neighborhood of C_{0} in ∂L_{0} ∩⟨F, O_{1}⟩. Let C′ ∈ ∂L∩F near C with corresponding projection C′_{0}. Setting C′ = (1 − t′)C′_{0} + t′O_{1} the condition C′ ∈ F reduces to ⟨C′ − O_{λ}, N⟩ = 0. Working this out in terms of t′ we obtain

t′ = 1 + \frac{1 − λ}{\frac{⟨C′_{0},N⟩}{⟨O_{1},N⟩} − 1}.  (24)

(Note that ⟨C′_{0}, N⟩ ≠ 0 as O_{1} ∉ F.) This formula holds for any type I, II, or III point C′. First assume that C is of type II. Since C′ is close to C, it is also a type II point. Thus

\frac{1}{1 + Λ(C′, O_{λ})} = \frac{λ}{t′}.

Comparing these formulas and recalling the assumption on Λ(., O_{λ}) restricted to F we see that, keeping N constant, C′_{0} ↦ ⟨C′_{0}, N⟩ attains a local maximum at C_{0}. The claim follows in this case. Now let C be a type III point. Since t^{∗} = t, comparing (19) and (24) (for t′ = t), we find

Λ(C_{0}, O_{0}) = −\frac{1}{λ} \frac{⟨C_{0}, N⟩}{⟨O_{1}, N⟩}.

Let C′ ∈ ∂L ∩ F be near C. By assumption

\frac{1}{1 + Λ(C, O_{λ})} ≤ \frac{1}{1 + Λ(C′, O_{λ})}.  (25)

Since we have no information about the type of C′ we split the study into two cases. First assume that C′ is a type I point. We rewrite (25) using Proposition 5 and (24) (twice) as

(1 − \frac{⟨C_{0}, N⟩}{⟨O_{1}, N⟩}) \frac{1}{1 + Λ(C_{0}, O_{0})} ≤ (1 − \frac{⟨C′_{0}, N⟩}{⟨O_{1}, N⟩}) \frac{1}{1 + Λ(C′_{0}, O_{0})}. (26)

In addition, since C′ is a type I point, with obvious notation, we also have t^{′∗} > t′. By (19) and (24), this gives

\frac{1}{1 + Λ(C′_{0}, O_{0})} < \frac{1}{1 − \frac{1}{λ} \frac{⟨C′_{0},N⟩}{⟨O_{1},N⟩}} .

Putting everything together in (26), we obtain

(1 − \frac{⟨C_{0}, N⟩}{⟨O_{1}, N⟩}) \frac{1}{1 − \frac{1}{λ} \frac{⟨C_{0},N⟩}{⟨O_{1},N⟩}} < (1 − \frac{⟨C′_{0}, N⟩}{⟨O_{1}, N⟩}) \frac{1}{1 − \frac{1}{λ} \frac{⟨C′_{0},N⟩}{⟨O_{1},N⟩}} .

This gives

⟨C_{0}, N⟩ < ⟨C′_{0}, N⟩  (27)

as stated. Assume now that C′ is a type II point. By Proposition 5, we have t ≥ t′ so that using (24), we again obtain (27). The claim follows. Note finally that since L_{0} ∩ ⟨F, O_{0}⟩ is convex, the local minimum at C is actually global. The above applies to all type II and type III configuration points. Let C_{i} be a type II or type III point. Let H_{0} ⊂ E_{0} ∩ ⟨F, O_{1}⟩ be the hyperplane that contains C_{i,0} and orthogonal to N. (The projection of N to E_{0} ∩ ⟨F, O_{1}⟩ is nonzero since otherwise F were parallel to E_{0} ∩ ⟨F, O_{1}⟩ and it would contain only type I configuration points.) Since C′ ↦ ⟨C′, N⟩ has a minimum at C_{i,0}, H_{0} is a supporting hyperplane to L_{0} ∩ ⟨F, O_{1}⟩. It follows that the projections (from O_{1} to L_{0}) of all the type II and type III configuration points are on H_{0}, and so is their convex hull. Thus the convex hull C of the type II and type III configuration points themselves is contained in ∂L. Taking opposites, we see that the opposite convex hull is contained in L_{0} ∩ ⟨F, O_{1}⟩. The distortion function Λ then must be linear on C. Since Λ has a local maximum at every vertex, it must be constant on C. We obtain that all type II and type III points can be compressed into a single point listed with multiplicity. This contradicts the minimality of m. Thus there is only one type II or type III configuration point. Finally, this cannot be a type II point since then its opposite must comprise the rest of the configuration points, and this is possible only for m = 1.

Summarizing, we see that in the configuration {C_{0}, …, C_{m}} there is a unique type III point and the rest of the points are contained in ∂L_{0}. We let C_{0} denote the type III point so that {C_{1}, …, C_{m}} ⊂ ∂L_{0}. We now set

C_{0} = (1 − t_{0})C_{0,0} + t_{0}O_{1}, 0 < t_{0} < 1, C_{0,0} ∈ ∂L_{0},  (28)

and consider the m-configuration {B_{0}, …, B_{m}} ∈ S_{m}^{0}(L_{0}, O_{0}), where B_{0} = C_{0,0} is the distinguished element and B_{i} = C_{i}, i = 1, …, m. We claim that

∑_{i=0}^{m} \frac{1}{1 + Λ(C_{i}, O_{λ})} = λ + (1 − λ) [\frac{1}{1 + Λ_{[B_{0},...,B_{m}]}(B_{0}, O_{0})} + ∑_{i=1}^{m} \frac{1}{1 + Λ_{L_{0}}(B_{i}, O_{0})}] .

(29)

With this, it will follow that

σ_{m}(L, O_{λ}) ≥ λ + (1 − λ)τ_{m}(L_{0}, O_{0}),  (30)

and the infimum in τ_{m}(L_{0}, O_{0}) is attained by a simplicial configuration. We first note that, by (20), for i = 1, …, m, we have

\frac{1}{1 + Λ(C_{i}, O_{λ})} = \frac{1 − λ}{1 + Λ_{L_{0}}(B_{i}, O_{λ})}.  (31)

Thus, to prove (29), using Proposition 5 and reverting to the C’s, we need to show that

\frac{1}{1 + Λ_{L}(C_{0}, O_{λ})} = λ + \frac{1 − λ}{1 + Λ_{[C_{0,0},...,C_{m}]}(C_{0,0}, O_{0})}.  (32)

We need to calculate the distortion at C_{0,0} with respect to the simplex [C_{0,0}, …, C_{m}]. To do this, we first determine the intersection of the line through C_{0,0} and O_{0} and the (m − 1)-simplex [C_{1}, …, C_{m}]. We have

∑_{i=0}^{m} μ_{i}C_{i} = O_{λ} and ∑_{i=0}^{m} μ_{i} = 1, 0 < μ_{i} < 1.

Expanding, we obtain

μ_{0}(1 − t_{0})C_{0,0} + ∑_{i=1}^{m} μ_{i}C_{i} = 0 and μ_{0}t_{0} = λ,

where we used that O_{0} is the origin. Since ∑_{i=1}^{m} μ_{i} = 1 − μ_{0} = 1 − λ/t_{0}, we have

−\frac{λ(1 − t_{0})}{t_{0} − λ}C_{0,0} = ∑_{i=1}^{m} \frac{μ_{i}}{1 − λ/t_{0}} C_{i} ∈ [C_{1}, …, C_{m}].

Thus, we have

Λ_{[C_{0,0},...,C_{m}]}(C_{0,0}, O_{0}) = \frac{t_{0} − λ}{λ(1 − t_{0})}.

With this, we obtain

λ + \frac{1 − λ}{1 + Λ_{[C_{0,0},...,C_{m}]}(C_{0,0}, O_{0})} = \frac{λ}{t_{0}} .

Now Proposition 6 gives (32). We now show that equality holds in (30). Let {B_{0}, …, B_{m}} ∈ S_{m}^{0}(L_{0}, O_{0}) be simplicial. We ‘lift’ this configuration to an m-configuration {C_{0}, …, C_{m}} ∈ C_{m}(L, O_{λ}) such that (29) holds with C_{0,0} = B_{0} and C_{i} = B_{i}, i = 1, …, m. To do this, we write ∑_{i=0}^{m} ν_{i}B_{i} = O_{0} with ∑_{i=0}^{m} ν_{i} = 1 and 0 < ν_{i} < 1. We define λ < t_{0} < 1 such that 1/t_{0}−1 = ν_{0}(1/λ−1), and then μ_{0} = λ/t_{0} and μ_{i} = (1−λ)ν_{i}, i = 1, …, m. We finally let C_{0} = (1 − t_{0})B_{0} + t_{0}O_{1} and C_{i} = B_{i}, i = 1, …, m. With these we have

∑_{i=0}^{m} μ_{i}C_{i} = \frac{λ}{t_{0}} C_{0} + (1 − λ) ∑_{i=1}^{m} ν_{i}C_{i}

= λ (\frac{1}{t_{0}} − 1) + λO_{1} + (1 − λ)(O_{0} − ν_{0}B_{0}) = O_{λ}.

In addition, we also have

∑_{i=0}^{m} μ_{i} = \frac{λ}{t_{0}} + (1 − λ) ∑_{i=1}^{m} ν_{i}
= \frac{λ}{t_{0}} + (1 − λ)(1 − ν_{0}) = 1.

These show that {C_{0}, …, C_{m}} ∈ C_{m}(L, O_{λ}). On the other hand, reversing the steps in the computation for (29) above, we obtain (29) again. With this, we have

λ + (1 − λ) [\frac{1}{1 + Λ_{[B_{0},...,B_{m}]}(B_{0}, O_{0})} + ∑_{i=1}^{m} \frac{1}{1 + Λ_{L_{0}}(B_{i}, O_{0})} ≥ σ_{m}(L_{0}, O_{0})] .

Taking the infimum, we arrive at

λ + (1 − λ)τ_{m}(L_{0}, O_{0}) ≥ σ_{m}(L_{0}, O_{0}).

Thus, equality holds in (30). Theorem F follows.

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Received June 30, 2007