A Spiky Ball

TL;DR

This paper demonstrates that in high-dimensional Euclidean space, there exist convex bodies very close to the sphere that require exponentially many translates to illuminate, highlighting complexity in covering convex shapes.

Contribution

It introduces a probabilistic method to show the existence of convex bodies near the Euclidean ball with exponentially large illumination numbers.

Findings

Existence of convex bodies close to the Euclidean ball with high illumination number

Illumination number grows exponentially with dimension for certain convex bodies

Probabilistic approach used to establish these results

Abstract

The Illumination Problem may be phrased as the problem of covering a convex body in Euclidean $n$-space by a minimum number of translates of its interior. By a probabilistic argument, we show that, arbitrarily close to the Euclidean ball, there is a centrally symmetric convex body of illumination number exponentially large in the dimension.