Minimizing the solid angle sum of orthogonal polyhedra

Abstract

In this paper we characterize the set of orthogonal polyhedra in R3 that minimize the sum of their internal solid angles. To prove our result, we first generalize the well known result that any orthogonal polygon in the plane with n vertices has (n+4)/2 convex and (n−4)/2 reflex vertices; see [1,2]. We prove that an orthogonal polyhedron of genus g with n vertices, k of which have degree greater than or equal to 4, has (n+8−8g+3k)/2 convex vertices and (n−8+8g−3k)/2 reflex vertices. We also prove that the sum of the solid angles of an orthogonal polyhedron is at least (n−4+4g)π, and at most (3n−24−4g)π. These inequalities are minimized and maximized, respectively, when g=0.