Inscribed and circumscribed radius of $\kappa$-convex hypersurfaces in Hadamard manifolds

TL;DR

This paper establishes bounds on the circumradius of convex polygons in Hadamard surfaces based on curvature constraints, contributing to geometric analysis in non-positive curvature spaces.

Contribution

It provides a new upper bound for the circumradius of convex polygons in Hadamard manifolds using curvature bounds at vertices.

Findings

Upper bound of circumradius in terms of curvature at vertices

Curvature constraints influence polygon size in Hadamard manifolds

Geometric bounds extend understanding of convex shapes in non-positive curvature spaces

Abstract

Let $P$ be a convex polygon in a Hadamard surface $M$ with curvature $K$ satisfying $-k_2^2 \ge K \ge -k_1^2$. We give an upper bound of the circumradius of $P$ in terms of a lower bound of the curvature of $P$ at its vertices.