An estimate for the anisotropic maximum curvature in the planar case

TL;DR

This paper establishes a lower bound for the anisotropic maximum curvature of smooth Jordan curves in the plane using anisotropic curvature flow, relating it to the enclosed area and the Wulff shape.

Contribution

It introduces a new estimate for anisotropic maximum curvature in the planar case, connecting curvature bounds with geometric quantities via anisotropic flow methods.

Findings

Proves a lower bound for anisotropic maximum curvature in terms of area and Wulff shape.

Uses anisotropic curvature flow to derive geometric inequalities.

Provides a link between curvature, area, and anisotropic norms in planar curves.

Abstract

We fix a Finsler norm $F$ and, using the anisotropic curvature flow, we prove that the anisotropic maximum curvature $k^F_{\max}$ of a smooth Jordan curve is such that $ k^F_{\max}(\gamma)\geq \sqrt{\kappa/A}$ , where $A$ is the area enclosed by $\gamma$ and $\kappa$ the area of the unitary Wulff shape associated to the anisotropy $F$.