Convex polygons in geometric triangulations

We show that the maximum number of convex polygons in a triangulation of n points in the plane is O(1.5029n). This improves an earlier bound of O(1.6181n) established by van Kreveld, Löffler, and Pach (2012) and almost matches the current best lower bound of Ω(1.5028n) due to the same authors. Given a planar straight-line graph G with n vertices, we show how to compute efficiently the number of convex polygons in G