A discrete Blaschke Theorem for convex polygons in $2$-dimensional space forms

Abstract

Let $M$ be a $2$-space form. Let $P$ be a convex polygon in $M$. For these polygons, we define (and justify) a curvature $\kappa_i$ at each vertex $A_i$ of the polygon and and prove the following Blaschke's type theorem: If $P$ is a convex plygon in $M$ with curvature at its vertices $\kappa_i\ge \kappa_0 >0$, then the circumradius $R$ of $P$ satisfies $ta_\lambda(R) \le \pi/(2\kappa_0)$ and the equality holds if and only if the polygon is a $2$-covered segment.