A Discrete Blaschke Theorem for Convex Polygons in 2-Dimensional Space Forms

Let M be a 2-dimensional 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 prove the following Blaschke-type theorem: "If P is a convex polygon in M with curvature at its vertices kappa(i) >= kappa(0) > 0, then the circumradius R of P satisfies ta(lambda) (R) <= pi /(2 kappa(0) ) and the equality holds if and only if the polygon is a doubly covered segment".