On the Stability of the p-Affine Isoperimetric Inequality

Abstract

Abstract. Employing the affine normal flow, we prove a stability version of the p-affine isoperimetric inequality for p ≥ 1 in R 2 in the class of origin-symmetric convex bodies. That is, if K is an origin-symmetric convex body in R 2 such that it has area π and its p-affine perimeter is close enough to the one of an ellipse with the same area, then, after applying a special linear transformation, K is close to an ellipse in the Hausdorff distance.

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).…”

“…Additionally, we can also control the distance between K and K s in the BanachMazur distance (using a Harnack estimate) provided that ε is small enough. Putting these observations altogether, we are able to prove that K 0 is close to the unit disk in the Banach-Mazur distance and so is K. This approach to the stability problems was employed by the author to obtain the stability of the p-affine isoperimetric inequality for bodies in K 2 e , [40].…”

“…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.…”

The spherical convex floating body

“…Affine surface area is among the most powerful tools in equiaffine differential geometry (see B. Andrews [5,6], A. Stancu [75,76], M. Ivaki [32] and M. Ivaki and A. Stancu [33]). It appears naturally as the Riemannian volume of a smooth convex hypersurface with respect to the affine metric (or Berwald-Blaschke metric), see e.g.…”

Stability of the Blaschke–Santaló inequality in the plane

“…Moreover, stability of the p-affine isoperimetric inequality also follows from the stability of the Blaschke-Santaló inequality (See [17,22] for definitions of the p-affine surface areas, and for the statements of the p-affine isoperimetric inequalities, and see also [13,14] for their generalizations in the context of the Orlicz-Brunn-Minkowski theory, basic properties, and affine isoperimetric inequalities they satisfy.). Stability of the p-affine isoperimetric inequality, in the Hausdorff distance, for bodies in K 2 e was established by the author in [12] via the affine normal flow with the order of approximation equal to 3/10. Therefore, the main theorem here replaces 3/10 by 1/2 and extends that result, if p > 1, to bodies with the Santaló points or centroids at the origin, and if p = 1, to any convex body in K 2 .…”