If You Can Hide behind It, Can You Hide inside It?

Let K and L be compact convex sets in Rn. Suppose that, for a given dimension 1 ≤ d ≤ n − 1, every d-dimensional orthogonal projection of L contains a translate of the corresponding projection of K. 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? A compact convex set L is defined to be d-decomposable if L is a direct Minkowski sum (affine Cartesian product) of two or more convex bodies each of dimension at most d. A compact convex set L is called d-reliable if, whenever each d-dimensional orthogonal projection of L contains a translate of the corresponding d-dimensional projection of K, it must follow that L contains a translate of K. It is shown that, for 1 ≤ d ≤ n− 1: (1) d-decomposability implies d-reliability. (2) A compact convex set L in Rn is d-reliable if and only if, for all m ≥ d+2, no m unit normals to regular boundary points of L form the outer unit normals of an (m− 1)-dimensional simplex. (3) Smooth convex bodies are not d-reliable. (4) A compact convex set L in Rn is 1-reliable if and only if L is 1-decomposable (i.e. a parallelotope). (5) A centrally symmetric compact convex set L in Rn is 2-reliable if and only if L is 2-decomposable. However, there are non-centered 2-reliable convex bodies that are not 2-decomposable. As a result of (5) above, the only reliable centrally symmetric covers in R3 from the perspective of 2-dimensional shadows are the affine convex cylinders (prisms). However, in dimensions greater than 3, it is shown that 3decomposability is only sufficient, and not necessary, for L to cover reliably with respect to 3-shadows, even when L is assumed to be centrally symmetric. Consider two compact convex subsetsK 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. 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? In dimension 2 it is easy to see that the answer is no. For example, if an equilateral triangle Δ is inscribed in a disc D of unit diameter, the slightly larger triangle (1 + )Δ still has less than unit width in every direction (provided > 0 is sufficiently small), but no longer fits inside D. The same construction works for any set K inscribed in D and having strictly less than unit diameter. Another Received by the editors June 4, 2009. 2000 Mathematics Subject Classification. Primary 52A20. c ©2011 Daniel A. Klain