Convex iso-Delaunay regions in strata of translation surfaces

Abstract

Strata of translation surfaces are covered by the closures of finitely many iso-Delaunay regions: open subspaces parametrizing surfaces whose Delaunay triangulations are combinatorially equivalent. We prove that the iso-Delaunay regions for triangulations with involutions called triangle matchings are convex with respect to the angle coordinates of the triangles. As a corollary, we show that all iso-Delaunay regions in hyperelliptic stratum components are convex. This work generalizes two directions in the theory of flat surfaces.