Convex polygons in geometric triangulations

We show that the maximum number of convex polygons in a triangulation of $n$ points in the plane is $O({1.5029}^n)$. This improves an earlier bound of $O({1.6181}^n)$ established by van Kreveld, L"offler, and Pach (2012) and almost matches the current best lower bound of $\mathrm{\Omega }({1.5028}^n)$ 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 $G$.