Moser's Shadow Problem

TL;DR

This paper determines the asymptotic bounds for Moser's shadow problem, revealing logarithmic growth rates for bounded and unbounded polyhedra, and extends previous work on silhouette span numbers.

Contribution

It provides the first asymptotic bounds for the shadow functions of bounded and unbounded polyhedra, completing the understanding of their growth rates.

Findings

Bounded shadow function $rak{s}_b(n)$ is $ heta(rac{ ext{log}(n)}{ ext{log}( ext{log}(n))})$.

Unbounded shadow function $rak{s}_u(n)$ is $ heta(1)$.

Unbounded silhouette span number $rak{s}_u^{ ext{*}}(n)$ grows as $ heta(rac{ ext{log}(n)}{ ext{log}( ext{log}(n))})$.

Abstract

Moser's shadow problem asks to estimate the shadow function $\mathfrak{s}_b(n)$, which is the largest number such that for each bounded convex polyhedron $P$ with $n$ vertices in $3$-space there is some direction ${\bf v}$ (depending on $P$) such that, when illuminated by parallel light rays from infinity in direction ${\bf v}$, the polyhedron casts a shadow having at least $\mathfrak{s}_b(n)$ vertices. A general version of the problem allows unbounded polyhedra as well, and has associated shadow function $\mathfrak{s}_u(n)$. This paper presents correct order of magnitude asymptotic bounds on these functions. The bounded case has answer $\mathfrak{s}_b(n) = \Theta \big( \log (n)/ (\log(\log (n))\big$. The unbounded shadow problem is shown to have the different asymptotic growth rate $\mathfrak{s}_u(n) = \Theta \big(1\big)$. Results on the bounded shadow problem follow from 1989 work of Chazelle, Edelsbrunner and Guibas on the (bounded) silhouette span number $\mathfrak{s}_b^{\ast}(n)$, defined analogously but with arbitrary light sources. We complete the picture by showing that the unbounded silhouette span number $\mathfrak{s}_u^{\ast}(n)$ grows as $\Theta \big( \log (n)/ (\log(\log (n))\big)$.