On the Multiple Illumination Numbers of Convex Bodies

In this paper, we introduce an m-fold illumination number Im(K)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I<^>m(K)$$\end{document} of a convex body K in Euclidean space Ed\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {E}<^>d$$\end{document}, 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)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I<^>m(K)$$\end{document} for any d-dimensional convex body K, and get an upper bound of Im(K)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I<^>m(K)$$\end{document} for any d-dimensional convex body K with smooth boundary. We also prove that Im(K)=2m+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I<^>m(K)=2m+1$$\end{document}, 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 Bd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {B}<^>d$$\end{document} in small dimensions. In particular, we show that , for a regular convex n-sided polygon P.