On the multiple illumination numbers of convex bodies

TL;DR

This paper introduces the concept of m-fold illumination numbers for convex bodies, establishing bounds and exact values in various dimensions and for specific shapes, advancing understanding of multi-directional illumination in convex geometry.

Contribution

It defines the m-fold illumination number, derives bounds for convex bodies, and computes exact values for specific cases like smooth bodies and polygons, extending classical illumination results.

Findings

Lower bound of I^m(K) for any convex body

Exact value of I^m(K) for 2D smooth convex bodies

Formula for I^m(P) for regular polygons

Abstract

In this paper, we introduce an $m$-fold illumination number $I^m(K)$ of a convex body $K$ in Euclidean space $\mathbb{E}^d$, which is the smallest number of directions required to $m$-fold illuminate $K$, i.e., each point on the boundary of $K$ is illuminated by at least $m$ directions. We get a lower bound of $I^m(K)$ for any $d$-dimensional convex body $K$, and get an upper bound of $I^m(\mathbb{B}^d)$, where $\mathbb{B}^d$ is a $d$-dimensional unit ball. We also prove that $I^m(K)=2m+1$, for a $2$-dimensional smooth convex body $K$. Furthermore, we obtain some results related to the $m$-fold illumination numbers of convex polygons and cap bodies of $\mathbb{B}^d$ in small dimensions. In particular, we show that $I^m(P)=\left\lceil mn/{\left\lfloor\frac{n-1}{2}\right\rfloor}\right\rceil$, for a regular convex $n$-sided polygon $P$.