On the Multiple Illumination Numbers of Convex Bodies

In this paper, we introduce an m-fold illumination number \(I^m(K)\) $I^m(K)$ of a convex body K in Euclidean space \(\mathbb {E}^d\) ${\mathbb{E}}^d$ , 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 \(I^m(K)\) $I^m(K)$ for any d-dimensional convex body K, and get an upper bound of \(I^m(K)\) $I^m(K)$ for any d-dimensional convex body K with smooth boundary. We also prove that \(I^m(K)=2m+1\) $I^m(K)=2m+1$ , 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 a d-dimensional unit ball \(\mathbb {B}^d\) ${\mathbb{B}}^d$ in small dimensions. In particular, we show that , for a regular convex n-sided polygon P.