An inequality for the maximum curvature through a geometric flow

TL;DR

This paper presents a new proof of an inequality relating maximum curvature and enclosed area of a smooth Jordan curve, utilizing the curve shortening flow to establish the result.

Contribution

The paper introduces a novel proof technique for the curvature-area inequality using the curve shortening flow, offering a new perspective on classical geometric inequalities.

Findings

Established the inequality $k_{max} \,\geq\, \sqrt{\pi/A}$ for smooth Jordan curves.

Demonstrated the effectiveness of the curve shortening flow in proving geometric inequalities.

Provided insights that could influence future research in geometric analysis.

Abstract

We provide a new proof of the following inequality: the maximum curvature $k_\mathrm{max}$ and the enclosed area $A$ of a smooth Jordan curve satisfy $k_\mathrm{max}\ge \sqrt{\pi/A}$. The feature of our proof is the use of the curve shortening flow.