Circumcenters in Finsler unitary groups

TL;DR

This paper investigates convexity and circumcenter existence in Finsler unitary groups using Schatten norms, leading to optimal bounds and applications in fixed point and rigidity problems.

Contribution

It introduces new convexity properties and proves the existence of circumcenters in Finsler unitary groups with Schatten norm metrics, extending previous results.

Findings

Existence of circumcenters for sets with radius less than π/2

Convexity properties of distance functions in Finsler unitary groups

Optimal bounds for convexity, circumcenter existence, and rigidity

Abstract

We study convexity properties of distance functions in Finsler unitary groups, where the Finsler structure is defined by translation of the $p$-Schatten norm on the Lie algebra. As a result we prove the existence of circumcenters for sets with radius less than $\pi/2$ in several metrics. This result is applied to a fixed point property and to quantitative metric bounds in certain rigidity problems. Bounds for convexity, existence of circumcenters and rigidity are shown to be optimal.