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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
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
Published electronically: April 11, 2011
MathSciNet review: 2806685
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Abstract: Let $ K$ and $ L$ be compact convex sets in $ \mathbb{R}^n$. Suppose that, for a given dimension $ 1 \leq d \leq 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 \leq d \leq n-1$:

(1)
$ d$-decomposability implies $ d$-reliability.

(2)
A compact convex set $ L$ in $ \mathbb{R}^n$ is $ d$-reliable if and only if, for all $ m \geq 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 $ \mathbb{R}^n$ is $ 1$-reliable if and only if $ L$ is $ 1$-decompos- able (i.e. a parallelotope).

(5)
A centrally symmetric compact convex set $ L$ in $ \mathbb{R}^n$ 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 $ \mathbb{R}^3$ 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 $ L$ to cover reliably with respect to $ 3$-shadows, even when $ L$ is assumed to be centrally symmetric.


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Additional Information

Daniel A. Klain
Affiliation: Department of Mathematical Sciences, University of Massachusetts Lowell, Lowell, Massachusetts 01854
Email: Daniel_Klain@uml.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-2011-05178-0
PII: S 0002-9947(2011)05178-0
Received by editor(s): June 4, 2009
Published electronically: April 11, 2011
Article copyright: © Copyright 2011 Daniel A. Klain