Circumcenters in Finsler unitary groups

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.

arXiv commentThis is an unpublished manuscript. The results in this manuscript were generalized to the infinite dimensional context in the preprint "Geometry of infinite dimensional groups: convexity and fixed points", arXiv:2203.06315, M. Miglioli