Nonlocal Flow of Convex Plane Curves and Isoperimetric Inequalities

2010/05/04 by Yu-Chu Lin, Lin, Yu-Chu, Dong-Ho Tsai +1
Mathematics · #35K15 #35K55 #Differential Geometry (math.DG) #FOS: Mathematics #math.DG #msc:35K15 #msc:35K55

paper · pdf · doi:10.48550/arxiv.1005.0438

23 pages

arxiv created 2010/05/04 · arxiv updated 2010/05/05

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.

Related

• A non-local area preserving curve flow
• Interpolation inequalities between the deviation of curvature and the isoperimetric ratio with applications to geometric flows
• The anisotropic polyharmonic curve flow for closed plane curves
• On a General Linear Nonlocal Curvature Flow of Convex Plane Curves
• On an area-preserving inverse curvature flow for plane curves