Containment and Inscribed Simplices
نویسنده
چکیده
Let K and L be compact convex sets in Rn. The following two statements are shown to be equivalent: (i) For every polytope Q ⊆ K having at most n+ 1 vertices, L contains a translate of Q. (ii) L contains a translate of K. Let 1 ≤ d ≤ n − 1. It is also shown that the following two statements are equivalent: (i) For every polytope Q ⊆ K having at most d+ 1 vertices, L contains a translate of Q. (ii) For every d-dimensional subspace ξ, the orthogonal projection Lξ of the set L contains a translate of the corresponding projection Kξ of the set K. It is then shown that, if K is a compact convex set in Rn having at least d + 2 exposed points, then there exists a compact convex set L such that every d-dimensional orthogonal projection Lξ contains a translate of the projection Kξ , while L does not contain a translate of K. In particular, if dimK > d, then there exists L such that every d-dimensional projection Lξ contains a translate of the projection Kξ , while L does not contain a translate of K. This note addresses questions related to following general problem: Consider two compact convex subsets K and L of n-dimensional Euclidean space. Suppose that, for a given dimension 1 ≤ d < n, every d-dimensional orthogonal projection (shadow) of L contains a translate of the corresponding projection of K. Under what conditions does it follow that the original set L contains a translate of K? In other words, if K can be translated to “hide behind” L from any perspective, does it follow that K can “hide inside” L? This question is easily answered when a sufficient degree of symmetry is imposed. For example, a support function argument implies that the answer is Yes if
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