On the multiple illumination numbers of convex bodies

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 $I^m(P)=\lceil mn\mathrm{/}\lfloor \frac{n-1}{2}\rfloor \rceil$, for a regular convex $n$-sided polygon $P$.