2023/06/23 by Kirati Sriamorn, Sriamorn, Kirati
Computer Science · Mathematics · #52A20 #52A55 #52C17 #Computational Geometry and Mesh Generation #FOS: Mathematics #Metric Geometry (math.MG) #Point processes and geometric inequalities
paper · pdf · doi:10.48550/arxiv.2306.13517
openalex publication_date 2023/06/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we introduce an m-fold illumination number Im(K) of a convex body K in Euclidean space 𝔼d, 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 Im(K) for any d-dimensional convex body K, and get an upper bound of Im(\mathbbBd), where \mathbbBd is a d-dimensional unit ball. We also prove that Im(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 \mathbbBd in small dimensions. In particular, we show that Im(P)=\lceil mn/\lfloor(n-1)/(2)\rfloor\rceil, for a regular convex n-sided polygon P.