Diffeomorphism groups of compact convex sets

TL;DR

This paper demonstrates that the group of smooth boundary-fixing diffeomorphisms of a compact convex set forms a C^0-regular infinite-dimensional Lie group, providing insights into differential equations on such sets.

Contribution

It establishes the Lie group structure and regularity of boundary-fixing diffeomorphisms on compact convex sets, a novel result in infinite-dimensional geometry.

Findings

G is a C^0-regular infinite-dimensional Lie group

Results on solutions to ODEs on compact convex sets

Enhanced understanding of diffeomorphism groups in finite-dimensional spaces

Abstract

For a compact convex subset K with non-empty interior in a finite-dimensional vector space, let G be the group of all smooth diffeomorphisms of K which fix the boundary of K pointwise. We show that G is a C^0-regular infinite-dimensional Lie group. As a byproduct, we obtain results concerning solutions to ordinary differential equations on compact convex sets.