Obtuse constants of Alexandrov spaces

TL;DR

This paper introduces the obtuse constant, a new geometric invariant for Alexandrov spaces with curvature bounds, exploring its relation to volume and establishing rigidity results for maximal values, with implications even in Riemannian geometry.

Contribution

The paper defines the obtuse constant for Alexandrov spaces and investigates its properties, including relations to volume and rigidity at maximal values, extending known results.

Findings

Relation between obtuse constant and normalized volume

Rigidity results for spaces with maximal obtuse constant

Results are new even in Riemannian geometry

Abstract

We introduce a new geometric invariant called the obtuse constant of spaces with curvature bounded below. We first find relations between this invariant and the normalized volume. We also discuss the case of maximal obtuse constant equal to $\pi/2$, where we prove some rigidity for spaces. Although we consider Alexandrov spaces with curvature bounded below, the results are new even in the Riemannian case.