Application of Andrews and Green-Osher inequalities to nonlocal flow of convex plane curves (Q1935529)

In this nice paper, the authors study a class of non-local flows on convex planar curves which keeps either the length of the curves, or their enclosed areas, constant. The normal speed is, pointwise, a difference of a function of the curvature at that point and a function of time (usually determined by the curves' length and/or area). A generalized Hölder inequality due to Andrews and a Green-Osher inequality (proved by Green and Osher via a flow) are exploited to show the convergence of these flows to circles. The key ingredient is to study the asymptotics of the isoperimetric difference of the evolving curves, \(L^2 (t) - 4 \pi A (t)\), where the above inequalities are used to show that it decays exponentially. Many examples of this class of flows are given and further applications of these inequalities are discussed.