Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 41 (2000), No. 2, 335-344.

A Geometric Proof of Some Inequalities Involving Mixed Volumes

Mathieu Meyer Shlomo Reisner

Equipe d’Analyse et de Mathématiques Appliquées, Université de Marne-la-Vallée 5, boulevard Descartes, Champs-sur-Marne, 77454 Marne-la-Vallée cedex 2, France meyer@math.univ-mlv.fr Department of Mathematics and School of Education - Oranim University of Haifa, Haifa 31905, Israel reisner@mathcs2.haifa.ac.il

Abstract. A new, “spherical harmonics free” proof of mixed-volume inequalities due to Schneider and to Goodey and Groemer, is presented.

Introduction The Alexandrov-Fenchel inequality implies that if K_{1} and K_{2} are two convex bodies in ℝ^{n}, and if for i, j = 1, 2, V_{ij} = V (K_{i}, K_{j}, B, …, B) is the mixed volume of K_{i}, K_{j} with n − 2 copies of the Euclidean unit ball B, then V_{12}^{2} ≥ V_{11}V_{22}. Moreover, it was proved by Schneider and by Goodey and Groemer that one can control the difference V_{12}^{2} − V_{11}V_{22} in the following way :

(1) If K_{2} = B, V_{12} = V_{1} = V (K_{1}, B, …, B), B_{1} is the Steiner ball of K_{1} (see the definitions below) and v_{n} is the volume of B then

V_{1}^{2} − v_{n}V_{11} ≥ \frac{n + 1}{n(n − 1)} v_{n} ∫_{S_{n−1}}(h_{K_{1}} − h_{B_{1}})^{2}dσ,

where σ denotes the surface measure on the sphere S_{n−1}, and for a convex body K, h_{K} denotes its support function.

(2) If the Steiner balls of K_{1}, K_{2} are the same, then

V_{12}^{2} − V_{11}V_{22} ≥ \frac{n + 1}{n(n − 1)} v_{n} ∫_{S_{n−1}}(h_{K_{1}} − h_{K_{2}})^{2}dσ.

0138-4821/93 $ 2.50 \begin{matrix}c \\ ◯ \end{matrix}2000 Heldermann Verlag

336 M. Meyer; S. Reisner: A Geometric Proof of Some Inequalities …

The results (1) and (2) may be interpreted as stability results in specific cases of the Alexandrov-Fenchel inequality. Using them one can derive further inequalities between intrinsic volumes of different orders of a given convex body. The proofs in [7] and [1] make use of spherical harmonics and of a representation of mixed volumes which involves the action of differential operators on support functions of convex bodies. We present here a new proof which is “spherical harmonics free” and which has a more geometric flavor. This proof is based on a variational argument involving Santaló’s inequality. We believe that this variational method may prove useful for the treatment of other problems as well. In fact we prove the following more general result (which also admits a proof using spherical harmonics):

(3) Let K_{1}, …, K_{p} be convex bodies in ℝ^{n} and for 1 ≤ i ≤ p, let V_{i} = V (K_{i}, B, …, B), h_{i} = h_{K_{i}} and let B_{i} be the Steiner ball of K_{i}. Then the quadratic form q : ℝ^{p} ↦ ℝ defined by q(s_{1}, …, s_{p}) = ∑_{i,j=1}^{p} a_{ij}s_{i}s_{j}, where

a_{ij} = V_{i}V_{j} − v_{n}V_{ij} − \frac{n + 1}{n(n − 1)} v_{n} ∫_{S_{n−1}}(h_{i} − h_{B_{i}})(h_{j} − h_{B_{j}}) dσ,

is non-negative.

The paper is organized as follows: after stating some lemmas, we give a new proof of (1), and then extend it to prove (3), of which (2) is an easy consequence.

Notations. Let K be a convex body in ℝ^{n}, endowed with its canonical scalar product ⟨ , ⟩. We denote by h_{K} the support function of K: for all x ∈ ℝ^{n}, h_{K}(x) = sup_{y∈K}⟨x, y⟩. If C is a Borel subset of ℝ^{n}, we denote by |C| its Lebesgue measure (its volume). We denote by S_{n−1} the unit sphere of the Euclidean space ℝ^{n}, endowed with its surface measure σ. If x_{0} ∈ K, the polar body K^{∗x_{0}} of K with respect to x_{0} is defined by

K^{∗x_{0}} = {y ∈ ℝ^{n}; ⟨y − x_{0}, x − x_{0}⟩ ≤ 1 for every x ∈ K}.

If 0 is in the interior of K, we write K^{∗} = K^{∗0}. We define (see [8]) the Steiner point z_{A} of a convex body A in ℝ^{n} by the identity

∫_{S_{n−1}}⟨x, u⟩ h_{A}(x) dσ(x) = v_{n} ⟨z_{A}, u⟩

for every u ∈ ℝ^{n}. The Steiner ball B(A) of the convex body A is

B(A) = z_{A} + \frac{w(A)}{2}B,

where w(A), the mean width of A, is defined by

w(A) = \frac{2}{nv_{n}} ∫_{S_{n−1}}h_{A}(x) dσ(x).

The Santaló point of K is the unique point x_{0} ∈ K with the property that x_{0} is the center of mass of K^{∗x_{0}}. It is well-known that this point is characterized by the fact that

|K^{∗x_{0}}| = min_{x∈K} |K^{∗x}| .

To say that x_{0} = 0 means that for every u ∈ ℝ^{n}, one has

∫_{K^{∗}}⟨x, u⟩ dx = 0.

We define now the volume product P(K) of K by

P(K) = min_{x∈ℝ^{n}} |K| · |K^{∗x}| = |K||K^{∗x_{0}}|.

By a celebrated theorem of Santaló (see [6], [4]), P(K) ≤ P(B) for all K.

Lemma 1. Let K, A be two convex bodies with 0 in their interiors. Let

C = {(y, t) ∈ ℝ^{n} × ℝ ; 1 + th_{A}(x) − ⟨x, y⟩ > 0 for all x ∈ K^{∗}},

and

D = {t ∈ ℝ ; (y, t) ∈ C for some y ∈ ℝ^{n}} .

Define F : C ↦ ℝ by

F (y, t) = ∫_{K^{∗}} \frac{1}{(1 + th_{A}(x) − ⟨x, y⟩)^{n+1}} dx.

Then there exists a > 0 satisfying ] − a, +∞[⊂ D and such that for each t > −a, the equation F (y, t) = min_{w;(w,t)∈C} F (w, t) has a unique solution y(t). Moreover, the mapping t ↦ y(t) is C^{∞} on ]− a, +∞[.

Proof. Observe first that C is an open convex subset of ℝ^{n}×ℝ, that 0 ∈ D and that F is C^{∞} and strictly convex on C (because u ↦ (1+u)^{−n−1} is strictly convex on ]−1, +∞[ ). By the proof of Lemma 2 below, for t ≥ 0 and (y, t) ∈ C, one has F (y, t) = |(K+tA)^{∗y}|. By [6] (see also Theorem 9 in [5] for a more precise statement and a complete proof), F (y, t) → +∞ when (y, t) → ∂C with fixed t ≥ 0. Thus by the strict convexity of F, for every t ≥ 0 there exists a unique y(t) such that (y(t), t) ∈ C and F (y(t), t) = min_{y;(y,t)∈C} F (y, t). Thus (∇_{y}F)(y(t), t) = 0. Define G : C ↦ ℝ^{n} by G(y, t) = (∇_{y}F)(y, t), and let D_{y}G be the differential of G at y ∈ ℝ^{n}. In the canonical basis of ℝ^{n}, the matrix of D_{y}G(y, t) is H_{y}F (y, t), the Hessian matrix of F as a function of y at (y, t). For any (y, t) ∈ C, one has

⟨w, H_{y}F (y, t)w⟩ = (n + 1)(n + 2) ∫_{K^{∗}} \frac{⟨x, w⟩^{2}}{(1 + th_{A}(x) − ⟨x, y⟩)^{n+3}} dx > 0 for w ∈ ℝ^{n} \ \{0\},

which implies det(H_{y}F (y, t)) > 0. Hence from the implicit functions theorem it follows that for t > 0, the unique solution y(t) of the equation G(y, t) = 0 is C^{∞} on ]0, +∞[.

We apply the same theorem in a neighbourhood of (y(0), 0) to get that for some a > 0 and some neighbourhood U of y(0) such that U×]− a, a[⊂ C, there exists a C^{∞} function z : ]− a, a[ ↦ U such that G(z(t), t) = 0. Since y(t) is the unique solution of F (y(t), t) = min_{y;(y,t)∈C} F (y, t) for t ≥ 0, it follows that y(t) = z(t) for t ≥ 0. Setting y(t) = z(t) for t ∈ ]− a, 0[, y(t) becomes a C^{∞} function on ]− a, +∞[. Moreover, again by the strict convexity of F on C, the equality G(y(t), t) = 0 implies that one has also F (y(t), t)) = min_{y;(y,t)∈C} F (y, t) for t ∈]− a, 0[. □

Lemma 2. Let K, A be two convex bodies with 0 in their interiors, let

E = {t ∈ ℝ; h_{K} + th_{A} is convex and positive on ℝ^{n} \ {0}},

and suppose that 0 is the Santaló point of K. For t ∈ E, denote by K + tA the convex body with support function h_{K} + th_{A}. Then, there exist y_{1}, y_{2} ∈ ℝ^{n} such that for all t ∈ E, the Santaló point y(t) of K + tA satisfies

y(t) = ty_{1} + t^{2}y_{2} + t^{2}ε(t) where ε(t) → 0 when t → 0, t ∈ E.

Moreover y_{1} is defined by the equality

∫_{K^{∗}}⟨x, u⟩h_{A}(x) dx = ∫_{K^{∗}}⟨x, u⟩⟨x, y_{1}⟩ dx for all u ∈ ℝ^{n}.

Proof. With the notations of Lemma 1, one has E ⊂ D. For t ∈ E and (y, t) ∈ C one has

|(K + tA)^{∗y}| = ∫_{K^{∗}} \frac{1}{(1 + th_{A}(x) − ⟨x, y⟩)^{n+1}} dx.

As a matter of fact, using polar coordinates,

∫_{K^{∗}} \frac{1}{(1 + th_{A}(x) − ⟨x, y⟩)^{n+1}} dx = ∫_{S_{n−1}}( ∫0^{\frac{1}{h_{K} (θ)}} \frac{r^{n−1}}{(1 + r(th_{A}(θ) − ⟨θ, y⟩))^{n+1}} dr) dθ

= \frac{1}{n} ∫_{S_{n−1}}[(\frac{r}{1 + r(th_{A}(θ) − ⟨θ, y⟩)})^{n} ]0^{\frac{1}{h_{K} (θ)}} dθ = \frac{1}{n} ∫_{S_{n−1}} \frac{1}{(h_{K}(θ) + th_{A}(θ) − ⟨θ, y⟩)^{n}} dθ

= \frac{1}{n} ∫_{S_{n−1}} \frac{1}{(h_{(K+tA−y)}(θ))^{n}} dθ = |(K + tA)^{∗y}| .

By Lemma 1, y(t) is C^{∞} on a neighbourhood of 0 (and actually also on E), hence Taylor’s formula of order 2 ensures the existence of the points y_{1} and y_{2} and of the function ε(t). Now, y(t) is characterized by the identity:

∫_{K^{∗}} \frac{⟨x, u⟩}{(1 + th_{A}(x) − ⟨x, y(t)⟩)^{n+2}} dx = 0 for every u ∈ ℝ^{n}

(this is exactly the fact that y(t) is the center of mass of (K+tA)^{∗y(t)}). Using the expansion of y(t), we get for every u ∈ ℝ^{n},

0 = ∫_{K^{∗}} \frac{⟨x, u⟩}{(1 + th_{A}(x) − ⟨x, ty_{1} + t^{2}y_{2} + t^{2}ε(t)⟩)^{n+2}} dx

= ∫_{K^{∗}}⟨x, u⟩ dx − (n + 2)t ∫_{K^{∗}}⟨x, u⟩(h_{A}(x) − ⟨x, y_{1}⟩) dx

+ t^{2}(n + 2) ∫_{K^{∗}} (⟨y_{2}, x⟩ + \frac{n + 3}{2}(h_{A}(x) − ⟨x, y_{1}⟩)^{2})⟨x, u⟩ dx + t^{2}ε(t).

Thus y_{1} satisfies

∫_{K^{∗}}⟨x, u⟩h_{A}(x) dx = ∫_{K^{∗}}⟨x, u⟩⟨x, y_{1}⟩ dx for all u ∈ ℝ^{n}

(clearly, only one point satisfies this equality). □

Lemma 3. With the assumptions and the notations of Lemma 2, let

|K + tA| = |K| + nV_{n−1,1}(K, A) t + \frac{n(n − 1)}{2}V_{n−2,2}(K, A)t^{2} + t^{2}η(t)

be the mixed volume expansion of |K + tA|, where η(t) → 0 when t → 0, t ≥ 0. Then

P(K + tA) = P(K) + c_{1}t + c_{2}t^{2} + t^{2}ε(t)

with ε(t) → 0 when t → 0, t ≥ 0 and

c_{1} = n|K^{∗}|V_{n−1,1}(K, A) − (n + 1)|K| ∫_{K^{∗}} h_{A}(x) dx,

c_{2} = \frac{n(n − 1)}{2}|K^{∗}|V_{n−2,2}(K, A) + \frac{(n + 1)(n + 2)}{2}|K| ∫_{K^{∗}}(h_{A}(x) − ⟨x, y_{1}⟩)^{2}dx

− n(n + 1)V_{n−1,1}(K, A) ∫_{K^{∗}} h_{A}(x) dx.

Proof. With the notations of Lemmas 1 and 2, for some functions ε such that ε(t) → 0 when t → 0, t ≥ 0, one has

|(K + tA)^{∗y(t)}| = ∫_{K^{∗}} \frac{1}{(1 + th_{A}(x) − ⟨x, y(t)⟩)^{n+1}} dx

= ∫_{K^{∗}} \frac{1}{(1 + (h_{A}(x) − ⟨x, y_{1}⟩)t − ⟨x, y_{2}⟩t^{2} + t^{2}ε(t))^{n+1}} dx

340 M. Meyer; S. Reisner: A Geometric Proof of Some Inequalities …

= |K^{∗}| − (n + 1)t ∫_{K^{∗}}(h_{A}(x) − ⟨x, y_{1}⟩) dx

− (n + 1)t^{2} ∫_{K^{∗}} (⟨x, y_{2}⟩ − \frac{n + 2}{2} (h_{A}(x) − ⟨x, y_{1}⟩)^{2}) dx + t^{2}ε(t).

Since by hypothesis ∫_{K^{∗}}⟨x, u⟩ dx = 0 for every u ∈ ℝ^{n}, we get |(K + tA)^{∗y(t)}| = |K^{∗}| − (n + 1)t∫_{K^{∗}}h_{A}(x) dx

+ \frac{(n + 1)(n + 2)}{2}t^{2} ∫_{K^{∗}}(h_{A}(x) −⟨x, y_{1}⟩)^{2}dx + t^{2}ε(t).

The result follows. □ The proof of the following lemma is easy. Lemma 4. Let K_{1} and K_{2} be convex bodies in ℝ^{n}, with Steiner balls B_{1} and B_{2} and Steiner points z_{K_{1}} and z_{K_{2}}. Then

∫_{S_{n−1}}(h_{K_{1}}(x) − h_{B_{1}}(x))(h_{K_{2}}(x) − h_{B_{2}}(x)) dσ(x)

= ∫_{S_{n−1}}(h_{K_{1}}(x) − ⟨z_{K_{1}}, x⟩)(h_{K_{2}}(x) − ⟨z_{K_{2}}, x⟩) dσ(x)

−\frac{1}{nv_{n}} ( ∫_{S_{n−1}}h_{K_{1}}(x) dσ(x))( ∫_{S_{n−1}}h_{K_{2}}(x) dσ(x)).

The following result was originally proved by Schneider [7] and by Goodey and Groemer [1]. Proposition 5. Let A be a convex body with Steiner ball B_{A}. Then

V_{1}(A)^{2} − v_{n}V_{2}(A) ≥ \frac{n + 1}{n(n − 1)} v_{n} ∫_{S_{n−1}}(h_{A}(x) − h_{B(A)}(x))^{2}dσ(x),

where V_{1}(A) = V (A, B, …, B) and V_{2}(A) = V (A, A, B, …, B) (B is the Euclidean unit ball). Proof. Let h = h_{A}, V_{1} = V_{1}(A) and V_{2} = V_{2}(A). By Santaló’s inequality, we know that

P(B + tA) ≤ P(B) = v_{n}^{2} for every t ≥ 0.

Since

n|B|V_{1} = v_{n} ∫_{S_{n−1}}h(x) dσ(x) = (n + 1)|B| ∫_{B} h(x) dx,

it follows from Lemma 3 that

\frac{n(n − 1)}{2}v_{n}V_{2} + \frac{(n + 1)(n + 2)}{2}v_{n} ∫_{B}(h(x) − ⟨x, y_{1}⟩)^{2}dx ≤ n(n + 1)V_{1} ∫_{B} h(x) dx,

or

\frac{n(n − 1)}{2}v_{n}V_{2} + \frac{n + 1}{2}v_{n} ∫_{S_{n−1}}(h(x) − ⟨x, y_{1}⟩)^{2}dσ(x) ≤ nV_{1} ∫_{S_{n−1}}h(x) dσ(x).

Observe that when K = B in Lemma 2, y_{1} is the Steiner point of A.

Since V_{1} = \frac{1}{n} ∫_{S_{n−1}}h(x) dσ, the preceding inequality is equivalent to

n^{2}V_{1}^{2} − \frac{n(n − 1)}{2}v_{n}V_{2} ≥ \frac{n + 1}{2}v_{n} ∫_{S_{n−1}}(h(x) − ⟨x, y_{1}⟩)^{2}dσ(x).

It follows from Lemma 4 that

n^{2}V_{1}^{2} − \frac{n(n − 1)}{2}v_{n}V_{2}

≥ \frac{n + 1}{2} v_{n}( ∫_{S_{n−1}}(h(x) − h_{B(A)}(x))^{2}dσ(x) + \frac{1}{nv_{n}} ( ∫_{S_{n−1}}h(x) dσ(x))^{2})

or equivalently

\frac{n(n − 1)}{2}(V_{1}^{2} − v_{n}V_{2}) ≥ \frac{n + 1}{2}v_{n} ∫_{S_{n−1}}(h(x) − h_{B(A)}(x))^{2}dσ(x).  □

We want now to generalize the method we used to more than one convex body. For this we use the so-called Minkowski differences or “érosions”. By the same arguments as in the proofs of Lemmas 2 and 3, one gets :

Lemma 6. Let K_{1}, …, K_{p} be convex bodies in ℝ^{n} with support functions h_{1}, …, h_{p}, and let

E = {s = (s_{1}, …, s_{p}) ∈ ℝ^{p} ; h_{B} + ∑_{i=1}^{p} s_{i}h_{i} is convex on ℝ^{n} and positive on ℝ^{n} \ {0}}.

For s ∈ E, let K(s) := B + ∑_{i=1}^{p} s_{i}K_{i} be the convex body whose support function is h_{B} + ∑_{i=1}^{p} s_{i}h_{i}. Then for s ∈ E, the Santaló point y(s) of K(s) satisfies

y(s) = ∑_{i=1}^{p} s_{i}y_{i} + ∑_{1≤i,j≤p} s_{i}s_{j}y_{ij} + ( ∑_{i=1}^{p} s_{i}^{2})ε(s),

where ε(s) → 0 when s → 0, s ∈ E, and for 1 ≤ i ≤ p, y_{i} is the Steiner point of K_{i} and y_{ij} ∈ ℝ^{n}, 1 ≤ i, j ≤ p. Moreover for s ∈ E,

|K(s)^{∗y(s)}|= |B| − (n + 1) ∑_{i=1}^{p} s_{i}∫_{B} h_{i}(x) dx

+\frac{(n + 1)(n + 2)}{2} ∑_{i,j=1}^{p}s_{i}s_{j} ∫_{B}(h_{i}(x) − ⟨x, y_{i}⟩)(h_{j}(x) − ⟨x, y_{j}⟩) dx + ( ∑_{i=1}^{p} s_{i}^{2})ε(s)

where ε(s) → 0 when s → 0, s ∈ E.

Lemma 7. With the notations of Lemma 6, assume that each K_{i} has a C^{2} boundary with positive curvature ; then there exists a > 0 such that ]− a, +∞[^{p}⊂ E. Moreover, if V_{i} = V (K_{i}, B, …, B) and V_{ij} = V (K_{i}, K_{j}, B, …, B), one has for all s ∈ ]− a, +∞[^{p},

|B + s_{1}K_{1} + … + s_{p}K_{p}| = |B| + n ∑_{i=1}^{p} V_{i}s_{i} + \frac{n(n − 1)}{2} ∑_{i,j=1}^{p} V_{ij}s_{i}s_{j} + ( ∑_{i=1}^{p} s_{i}^{2}) ε(s),

where ε(s) → 0 when s → 0.

Proof. For the first fact, see [8], Theorem 2.5.4. The second part follows from the extension of mixed volumes to n-tuples of differences of support functions (see [8], Section 5.2, or another proof in [3]). □

Now using Lemmas 6 and 7 and the same calculations as in Lemma 3, observing that nv_{n}V_{i} = (n + 1)v_{n} ∫_{B} h_{i}(x) dx, we get

Lemma 8. With the notations of Lemma 7,

P(K(s)) = P(B) + ∑_{i,j=1}^{p} c_{ij}s_{i}s_{j} + (∑_{i=1}^{p} s_{i}^{2})ε(s),

where ε(s) → 0 when s → 0 and for 1 ≤ i, j ≤ p,

c_{ij} = \frac{n(n − 1)}{2} v_{n}V_{ij} + \frac{(n + 1)(n + 2)}{2} v_{n} ∫_{B}(h_{i}(x) − ⟨x, y_{i}⟩)(h_{j}(x) − ⟨x, y_{j}⟩) dx

− n(n + 1)V_{i} ∫_{B} h_{j}(x) dx.

Theorem. Let K_{1}, …, K_{p} be convex bodies in ℝ^{n} and let B be the Euclidean ball. For 1 ≤ i, j ≤ p, let V_{i} = V (K_{i}, B, …, B), V_{ij} = V (K_{i}, K_{j}, B, …, B), h_{i} = h_{K_{i}}, let B_{i} be the Steiner ball of K_{i} and

a_{ij} = V_{i}V_{j} − v_{n}V_{ij} − \frac{n + 1}{n(n − 1)} v_{n} ∫_{S_{n−1}}(h_{i}(x) − h_{B_{i}}(x))(h_{j}(x) − h_{B_{j}}(x)) dσ(x).

Then the quadratic form q : ℝ^{p} ↦ ℝ defined by q(s_{1}, …, s_{p}) = ∑_{i,j=1}^{p} a_{ij}s_{i}s_{j} is nonnegative.

Proof. We approximate K_{i} by bodies having C^{2} boundaries with positive curvature. Using homogeneity, we may assume, due to Lemma 7, that h_{B} +∑_{i=1}^{p} s_{i}h_{i} is the support function of a convex body K(s) for all s = (s_{1}, …, s_{p}) ∈ [−1, 1]^{p}. Then by Santaló’s inequality, P(K(s)) ≤ P(B). The result follows now from Lemmas 4 and 8, with the same proof as in Proposition 5 (we needed here the fact that q(s) ≥ 0 for s in a neighborhood of 0, this is what necessitated the introduction of Lemma 7, that is the extension of Steiner’s formula to negative values of s_{i}). □

Remark. Note that Proposition 5 is a corollary of the theorem, since the latter implies a_{ii} ≥ 0.

Corollary 9. ([7], [1]) If the Steiner points and the mean widths of K_{1} and K_{2} are the same, then

2V_{12} − V_{11} − V_{22} ≥ \frac{n + 1}{n(n − 1)} ∫_{S_{n−1}}(h_{1}(x) − h_{2}(x))^{2}dσ(x).

Hence, for any K_{1} and K_{2} one has

V_{12}^{2} − V_{11}V_{22} ≥ \frac{n + 1}{n(n − 1)} w(K_{1})^{2}V_{22} ∫_{S_{n−1}}(hK̃_{1}(x) − hK̃_{2}(x))^{2}dσ(x)

where K̃_{1}, K̃_{2} are homothetic copies of K_{1}, K_{2}, with w(K̃_{1}) = w(K̃_{2}) = 1 and zK̃_{1} = zK̃_{2}.

Proof. The second assertion is an easy consequence of the first one (see [7] and [1]). To prove the first assertion, one may assume that B is the common Steiner ball of K_{1} and K_{2}. Then, with p = 2 in the theorem,

a_{12} = v_{n}^{2} − v_{n}V_{12} − \frac{n + 1}{n(n − 1)} v_{n} ∫_{S_{n−1}}(h_{1}(x) − 1)(h_{2}(x) − 1) dσ(x).

From the non-negativity of the quadratic form we get a_{11} + a_{22} ≥ 2\sqrt{a_{11}a_{22}} ≥ 2a_{12}, which is the first assertion. □

References

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[2] Groemer, H.: Geometric applications of Fourier series and spherical harmonics. Encyclopedia Math. Appl. 61, Cambridge Univ. Press, Cambridge 1996.

[3] Matheron, G.: La formule de Steiner pour les érosions. J. Appl. Probab. 15 (1978), 126–135.

[4] Meyer, M.; Pajor, A.: On the Blaschke-Santaló inequality. Arch. Math. 55 (1990), 82–93.

[5] Meyer, M.; Werner, E.: The Santaló-regions of a convex body. Trans. Amer. Math. Soc. 350 (1998), 4569–4591.

[6] Santaló, L. A.: Un invariante afin para los cuerpos convexos del espacio de n dimensiones. Portugal. Math. 8 (1949), 155–161.

[7] Schneider, R.: Stability in the Aleksandrov-Fenchel-Jessen theorem. Mathematika 36 (1989), 50–59.

[8] Schneider, R.: Convex bodies: The Brunn-Minkowski Theory. Encyclopedia Math. Appl. 44, Cambridge Univ. Press, Cambridge 1993.

Received April 13, 1999; revised version August 16, 1999