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

2023/05/12 by Alexander Borisenko, Borisenko, Alexander, Vicente Miquel +1
Mathematics · Computer Science · #Point processes and geometric inequalities #Computational Geometry and Mesh Generation #Geometric and Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2305.07566

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 κ_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 κ_i≥ κ₀ >0, then the circumradius R of P satisfies ta_λ(R) ≤ π/(2κ₀) and the equality holds if and only if the polygon is a 2-covered segment.

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