So, I was judging a science fair the other day, and one of the entrants was essentially this [ http://math.mit.edu/research/highschool/primes/materials/2016/Cohen-Rowley.p... ], describing a discrete analog of curve shortening flow, where points on a closed curve are moved normal to the curve and proportional to their curvature, with the [known] theorem being that the curves tend toward a circle and shrink to a point. The discrete version just moves the vertices of a polygon. The discrete analog they chose for curvature was pi - alpha, where alpha is the vertex angle, and they showed that the result, starting with a triangle, was either a line or a point, depending on certain conditions about the triangle. It [almost] immediately occurred to me that, for triangles, if you used the distance to the incenter as the curvature, the triangle would remain similar to itself and shrink to a point. Essentially, the "curvature" would be csc alpha/2. And, in fact, this would work for convex polygons as well. But of course, an asymmetric polygon would remain self-similar and so asymmetric. So, is there a discrete analog of curvature which would cause a polygon to "circularize" as it shrunk to a point, and if so, is it in any sense unique? Maybe something like (csc alpha/2)**2