A Fenchel Theorem for the Gauss maps and uniqueness of minimizers of nonlocal curvature energies

TL;DR

This paper establishes a Fenchel theorem for Gauss maps, demonstrating that circles and disks uniquely minimize certain nonlocal curvature energies through geometric analysis techniques.

Contribution

It introduces a Fenchel theorem for Gauss maps and proves the minimality of circles and disks for various generalized curvature energies.

Findings

Circles minimize most generalized tangent-point energies.

Disks minimize all fractional Willmore energies among convex planar sets.

Sharp lower bounds for the path length of Gauss maps are provided.

Abstract

In this paper, we prove a Fenchel theorem for Gauss maps by providing sharp lower bounds for the path length of Gauss maps of an embedding. By combining the Fenchel-type theorem with various techniques from the field of geometric analysis, we show that circles minimize most generalized tangent-point energies. Furthermore, we prove that disks minimize all fractional Willmore energies among the class of convex planar sets.