How many potatoes are in a mesh?

Abstract

We consider the combinatorial question of how many convex polygons can be made at most by using the edges taken from a fixed triangulation of n vertices. For general triangulations, there can be exponentially many: Ω(1.5028 n) and O(1.62n) in the worst case. If the triangulation is fat (every triangle has its angles lower-bounded by a constant δ > 0), then there can be only polynomially many: Ω(n 1/2 ⌊ 2π/δ ⌋ and O(n ⌈π/δ⌉). If we count convex polygons with the additional property that they contain no vertices of the triangulation in their interiors, we get the same exponential bounds in general triangulations, and Ω(n⌊2π/3δ⌋) and O(n ⌊2π/3δ⌋) in fat triangulations. © Springer-Verlag 2012.