ISSN 1088-6850(online) ISSN 0002-9947(print)

If you can hide behind it, can you hide inside it?

Author: Daniel A. Klain
Journal: Trans. Amer. Math. Soc. 363 (2011), 4585-4601
MSC (2000): Primary 52A20
Posted: April 11, 2011
MathSciNet review: 2806685
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Abstract: Let and be compact convex sets in . Suppose that, for a given dimension , every -dimensional orthogonal projection of contains a translate of the corresponding projection of . Does it follow that the original set contains a translate of ? In other words, if can be translated to hide behind'' from any perspective, does it follow that can hide inside'' ?

A compact convex set is defined to be -decomposable if is a direct Minkowski sum (affine Cartesian product) of two or more convex bodies each of dimension at most . A compact convex set is called -reliable if, whenever each -dimensional orthogonal projection of contains a translate of the corresponding -dimensional projection of , it must follow that contains a translate of .

It is shown that, for :

(1)
-decomposability implies -reliability.

(2)
A compact convex set in is -reliable if and only if, for all , no unit normals to regular boundary points of form the outer unit normals of an -dimensional simplex.

(3)
Smooth convex bodies are not -reliable.

(4)
A compact convex set in is -reliable if and only if is -decompos- able (i.e. a parallelotope).

(5)
A centrally symmetric compact convex set in is -reliable if and only if is -decomposable.

However, there are non-centered -reliable convex bodies that are not -decomposable.

As a result of (5) above, the only reliable centrally symmetric covers in from the perspective of 2-dimensional shadows are the affine convex cylinders (prisms). However, in dimensions greater than 3, it is shown that 3-decomposability is only sufficient, and not necessary, for to cover reliably with respect to -shadows, even when is assumed to be centrally symmetric.

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