Moser’s shadow problem

  • Jeffrey C. Lagarias

    University of Michigan, Ann Arbor, USA
  • Yusheng Luo

    Harvard University, Cambridge, USA
  • Arnau Padrol

    Sorbonne Université, Paris, France

Abstract

Moser's shadow problem asks to estimate the shadow function , which is the largest number such that for each bounded convex polyhedron with vertices in 3-space there is some direction (depending on ) such that, when illuminated by parallel light rays from infinity in direction , the polyhedron casts a shadow having at least vertices. A general version of the problem allows unbounded polyhedra as well, and has associated shadow function . This paper presents correct order of magnitude asymptotic bounds on these functions. The bounded shadow problem has answer The unbounded shadow problem is shown to have the different asymptotic growth rate . Results on the bounded shadow problem follow from 1989 work of Chazelle, Edelsbrunner and Guibas on the (bounded) silhouette span number , defined analogously but with arbitrary light sources. We complete the picture by showing that the unbounded silhouette span number grows as .

Cite this article

Jeffrey C. Lagarias, Yusheng Luo, Arnau Padrol, Moser’s shadow problem. Enseign. Math. 64 (2018), no. 3/4, pp. 477–496

DOI 10.4171/LEM/64-3/4-12