New four-vertex type theorems for spherical polygons

Abstract

We prove discrete analogs of four-vertex type theorems of spherical curves, which imply corresponding results for space polygons. The smooth theory goes back to the work of Beniamino Segre and, more recently, by Mohammad Ghomi, and consists of theorems that state, for a given closed spherical curve, nontrivial lower bounds on the number of spherical inflections plus the number of self and/or antipodal intersections counted with multiplicity. We study these concepts and results adapted to the case of spherical polygons and prove, using only discrete tools, the corresponding theorems.