If you can hide behind it, can you hide inside it?
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- by Daniel A. Klain PDF
- Trans. Amer. Math. Soc. 363 (2011), 4585-4601
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 $1 \leq d \leq n-1$:
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[(1)] $d$-decomposability implies $d$-reliability.
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[(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.
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[(3)] Smooth convex bodies are not $d$-reliable.
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[(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).
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[(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
- Received by editor(s): June 4, 2009
- Published electronically: April 11, 2011
- © Copyright 2011 Daniel A. Klain
- Journal: Trans. Amer. Math. Soc. 363 (2011), 4585-4601
- MSC (2000): Primary 52A20
- DOI: https://doi.org/10.1090/S0002-9947-2011-05178-0
- MathSciNet review: 2806685