Centro–affine curvature flows on centrally symmetric convex curves

Abstract

Abstract. We consider two types of p-centro affine flows on smooth, centrally symmetric, closed convex planar curves, p-contracting, respectively, pexpanding. Here p is an arbitrary real number greater than 1. We show that, under any p-contracting flow, the evolving curves shrink to a point in finite time and the only homothetic solutions of the flow are ellipses centered at the origin. Furthermore, the normalized curves with enclosed area π converge, in the Hausdorff metric, to the unit circle modulo SL(2). As a p-expanding flow is, in a certain way, dual to a contracting one, we prove that, under any p-expanding flow, curves expand to infinity in finite time, while the only homothetic solutions of the flow are ellipses centered at the origin. If the curves are normalized as to enclose constant area π, they display the same asymptotic behavior as the first type flow and converge, in the Hausdorff metric, and up to SL(2) transformations, to the unit circle. At the end, we present a new proof of p-affine isoperimetric inequality, p ≥ 1, for smooth, centrally symmetric convex bodies in R 2 .

The planar Busemann-Petty centroid inequality and its stability

“…We will state several lemmas from [6,38,40,42] to prepare the proof of Theorem B. The first lemma is rewriting identity (4.2).…”

“…A proof of the first part of the claim is given in [38]. To prove the second part of the claim, we may first apply a special linear transformation Φ ∈ SL(2) such that ΦE out is a disk.…”

“…The long time behavior of the flow in R n with K 0 ∈ F n e was investigated in [38,44,76]. It was proved that for 1 < p < n n−2 the volume-preserving p-flow, which keeps the volume of the evolving bodies xed and equal to the volume of the unit ball, evolves each body in F n e to the unit ball in C ∞ , modulo SL(n).…”

“…It was proved that for 1 < p < n n−2 the volume-preserving p-flow, which keeps the volume of the evolving bodies xed and equal to the volume of the unit ball, evolves each body in F n e to the unit ball in C ∞ , modulo SL(n). Two applications arising from the tools developed in [38] to the L −2 Minkowski problem and to the stability of the p-affine isoperimetric inequality in R 2 were given in [39,40]. The case p = 1, corresponds to the well-known affine normal flow.…”

“…It is, probably, worth mentioning that the method to conclude the long time behavior in this paper is significantly different from the method of [38] and in particular relies on a number of affinely associated bodies from convex geometry, namely K * , ΠK, and ΛK (See the next section for the definitions of these bodies.). Essentially, monotonicity of V (ΓK t )/V (K t ) along flow (1.5) plays a key role.…”