Illuminating Primitive Polytopes

Abstract

A convex $d$-dimensional polytope may be defined as a bounded intersection of closed halfspaces in $\mathbb{E}^d$ with interior. A polytope is primitive if omitting any halfspace renders the intersection unbounded. In this paper, we prove the Illumination Conjecture for the primitive polytopes: any primitive convex $d$-polytope $P$ can be illuminated with at most $2^d$ directions; even fewer if $P$ is not a linear image of a $d$-cube.