Convex bodies in Euclidean and Weil-Petersson geometries

TL;DR

This paper introduces a new variational formulation for the Funk metric on convex bodies and generalizes the Weil-Petersson metric on Teichmüller spaces, highlighting similarities with Thurston's asymmetric metric.

Contribution

It presents a novel variational approach to the Funk metric and extends the Weil-Petersson metric to new geometric contexts, revealing shared properties with Thurston's metric.

Findings

New variational formulation for Funk metric

Generalization of Weil-Petersson metric

Shared properties with Thurston's asymmetric metric

Abstract

On a convex body in a Euclidean space, we introduce a new variational formulation for its Funk metric, a Finsler metric compatible with the tautological Finsler structure of the convex body. We generalize the metric on Teichmuller spaces with the Weil-Petersson distance function. A set of similarities the resulting metric structure shares with Thurston's asymmetric metric is noted.