Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 38 (1997), No. 1, 87-90.

On Point Sets fixing a Convex Body from within

Peter Braß Irmina Herburt

Institut für Mathematik, Universität Greifswald D-17487 Greifswald, Germany Institute of Mathematics, Warsaw University of Technology Pl. Politechniki, 00-661 Warsaw, Poland

Abstract: We study the concept of a set fixing a convex body from within that was recently proposed by V. Soltan. We prove his conjecture that a finite set which fixes a d-dimensional convex body from within contains a subset of at most 2d points with the same property.

1. Introduction

The classical definition [2],[4] of a set S fixing a convex body K ⊂ ℝ^{d} is that S ⊂ K and for each translation by a vector t ∈ ℝ^{d}, t ≠ 0 there is a point s ∈ S and an ε > 0 such that s ∉ interior K but s ∈ interior(K + εt). This models a solid object K which is held in its place (against translations) by some points S outside K which would have to move inside K if one tried to force a movement. At the 1995 Intuitive Geometry Conference, V. Soltan defined a new concept of a fixing set. He calls a set S fixing the convex body K ⊂ ℝ^{d} from within if S ⊂ K and for each translation t ≠ 0 there is an s ∈ S such that s ∉ K + t. This models a rigid surface ∂K held in its place by some points S inside K which would have to penetrate that surface if one tried to force a movement of K.

Special interest found those fixing sets that are minimal with respect to inclusion, called primitive fixing systems. It was proved by Tomor [8] that in dimension 2 a primitive fixing system contains at most six points (see also [3]). For dimensions d ≥ 3 Bollobás constructed arbitrary large (finite) primitive fixing systems [1].

For the new concept V. Soltan conjectured [6] that in ℝ^{d} any finite set of points fixing a d-dimensional body from within has a subset of at most 2d points with the same property. We prove a slightly stronger theorem.

Theorem. Let K ⊂ ℝ^{d} be a convex body, U ⊂ ℝ^{d} a linear subspace and S ⊂ K a finite set with the property that for all u ∈ U, u ≠ 0, there is an s ∈ S such that s ∉ K + u. Then there is a subset T ⊆ S of at most 2 dim U elements with the same property.

0138-4821/93 $ 2.50 \begin{matrix}c \\ ◯ \end{matrix}1997 Heldermann Verlag

2. Proof of the Theorem

We first need the following lemmas, the first of which follows from Theorem 14.6 of [5].

Lemma 1. If C is a closed cone which does not contain a nontrivial linear subspace, then there is a t such that (t, c) < 0 for all c ∈ C, c ≠ 0.

Lemma 2. If C is a cone generated by X, and W is the maximum linear subspace contained in C, then W is generated as a cone by W ∩ X.

Proof of Lemma 2. Let V be the linear space that contains C, so W ⊆ V. The proof is by induction on dim(V) − dim(W), the case dim(V) = dim(W) being trivial. If V ≠ W, then there is a supporting hyperplane H of C at its apex 0. Then W ⊂ H ∩ C, and H ∩ C is a cone in H that is generated by X ∩ H. Since dim(H) < dim(V), we may apply the inductive assumption on H ∩ C, H ∩ X and W. To prove now the theorem, we may assume, since S is finite, that S is minimal with respect to inclusion. So S does not contain points s ∈ S that lie in the relative interior of the cut K ∩ (U + s) generated by them. Therefore N(s) := {n ∈ U | (n, v) ≥ 0 for all v ∈ (K−s)∩U} contains for each s at least one element (at least one normal to each supporting hyperplane (in U + s) of K ∩ (U + s) in s; if K ∩ (U + s) is fulldimensional in U + s, then the interior normal.).

The proof is by induction on dim U. The case dim U = 1 is trivial. Let dim U = i > 1. Consider the cone C ⊆ U generated by ⋃_{s∈S} N(s), i.e. the set of all nonnegative linear combinations of elements of ⋃_{s∈S} N(s). Suppose C does not contain a linear subspace. Since C is a closed cone (all the sets N(s) are closed) by Lemma 1 there is a vector t ∈ U such that (t, v) < 0 for all v ∈ C, v ≠ 0. Then the movement by t cannot be blocked by S. Conversely, assume that s ∉ K +ϵt for all positive ϵ, then there is a supporting hyperplane (in U + s) of K ∩ (U + s) in s which (weakly) separates the halfline s − ϵt (ϵ > 0) from K ∩ (U + s). The interior normal of that hyperplane has nonnegative inner product with t and belongs to N(s), contrary to the assumption.

So C does contain some linear subspace. Let W ⊆ C ⊆ U be the maximal linear subspace contained in C. Then by Lemma 2 this subspace W is generated as a cone by (⋃_{s∈S} N(s)) ∩ W. From this set (⋃_{s∈S} N(s)) ∩ W we select (using Steinitz’s theorem) a subset of at most 2 dim W elements that generates W (as a cone). Now this subset blocks all movement apart that in directions orthogonal to W. To see this, assume that t is a direction that has a nonzero inner product with some element of W; then (W = −W) it has positive inner product with some element of W. If a vector has a positive inner product with an element of a cone, it has a positive inner product with one of the generators of the cone, so t has positive inner product with one of the selected generators of W, which is an interior normal of a supporting hyperplane at some point s ∈ S, (one of the at most 2 dim W points that correspond to the selected generators of W). This point s blocks the movement in direction t.

So we only have to block movement in the directions of the orthogonal complement of W in U. This is done by the inductive assumption. This completes the proof.

3. Infinite Counterexamples

This theorem fails for infinite S, as can be seen by taking K := [0, 1] and S :=]0, 1[. In fact, for any K there are infinite S for which the theorem fails: We only have to take two strictly antipodal points a and b (so K lies between two parallel hyperplanes which touch K only at a and b). Then {a, b} fixes K. If we now replace b by a sequence of points of K converging to b (either from within, or along the boundary following one direction), we obtain a set that fixes K but does not contain a finite subset which fixes K.

Surprisingly, it can even fail for a closed set S, as can be seen by the following 3dimensional example of Soltan [7]: Let p_{n} := (sin \frac{π}{n}, cos \frac{π}{n}, 0) and q_{n} := (sin \frac{π}{n}, cos \frac{π}{n}, \frac{1}{n}), and let p_{0} := (0, 1, 0), q_{0} := (0, 1, 1). Then S := {p_{0}, p_{1}, p_{2}, …} is a closed set that fixes K := conv(⋃_{i=0}^{∞}{p_{i}, q_{i}}) from within, but no finite subset of S does (each set containing p_{0}, p_{1} and one further of the p_{i} blocks each movement apart that in direction (0, 0, −1), but no finite subset of S blocks movement in that direction).

Figure 1

In none of these infinite counterexamples the fixing set S is inclusion-minimal.

References

[1] Bollobás, B.: Fixing system for convex bodies. Studia Sci. Math. Hungar. 2 (1967), 351–354.

[2] Fejes Tóth, L.: On primitive polyhedra. Acta Math. Hungar. 13 (1962), 379–382.

[3] Fudali, S.: Six-point primitive fixing systems in the plane. Demonstratio Math. 19 (1986), 341–348.

[4] Grünbaum, B.: Fixing systems and inner illumination. Acta Math. Hungar. 15 (1964), 161–163.

[5] Rockafellar, R. Tyrell: Convex Analysis. Princeton Mathematical Series 28, Princeton Univ. Press 1970.

[6] Soltan, V.: Problem at the Problem Session of Intuitive Geometry 1995. to appear in Bolyai Soc. Math. Studies 90 P.Braß, I. Herburt: On point sets fixing a convex body from within

[7] Soltan, V.: Personal Communication 1995

[8] Tomor, B.: The fixing problem for convex figures (Hungarian). Mat. Lapok 14 (1963), 120–123.

Received January 8, 1996