Convex drawings of the complete graph: topology meets geometry

In a geometric drawing of Kn, trivially each 3-cycle bounds convex region: if two vertices are in that region, then so is the (geometric) edge between them. We define topological D Kn to be closed region R such any have (topological) them contained R.While drawings generalize drawings, they specialize ones. Therefore it might surprising all optimal (that is, crossing-minimal) were convex. However, we take first step showing convex: show has non-convex K5 whose extensions K7 no other K5, not (without reference conjecture for crossing number Kn). This example non-trivial local considerations providing sufficient conditions suboptimality. At our request, Aichholzer computationally verified that, up n = 12, every convex.Convexity naturally lends itself refinements, including hereditarily (h-convex) and face (f-convex). The hierarchy rectilinear ? f-convex h-convex provides links drawings. It known equivalent pseudolinear (generalizing rectilinear) pseudospherical spherical geodesic). characterize h-convexity by three forbidden subdrawings.This framework consider generalizations questions point sets plane. provide examples questions, namely numbers empty triangles existence k-gons.