Nonlocal Flow of Convex Plane Curves and Isoperimetric Inequalities

TL;DR

This paper surveys nonlocal flows of convex plane curves, introduces a new flow, and explores conditions for curves to be homothetic or parallel based on isoperimetric measures.

Contribution

It provides a comprehensive review of existing nonlocal flows, introduces a novel flow, and establishes new criteria for curve similarity and parallelism.

Findings

Analysis of properties related to area and length during flows

Introduction of a new nonlocal flow and its evolution behavior

Conditions for homothety and parallelism based on isoperimetric ratios

Abstract

In the first part of the paper we survey some nonlocal flows of convex plane curves ever studied so far and discuss properties of the flows related to enclosed area and length, especially the isoperimetric ratio and the isoperimetric difference. We also study a new nonlocal flow of convex plane curves and discuss its evolution behavior. In the second part of the paper we discuss necessary and sufficient conditions (in terms of the (mixed) isoperimetric ratio or (mixed) isoperimetric difference) for two convex closed curves to be homothetic or parallel.