New approach to the affine Pólya–Szegö principle and the stability version of the affine Sobolev inequality

Abstract

Inspired by a recent work of Haddad, Jiménez and Montenegro, we give a new and simple approach to the recently established general affine Pólya-Szegö principle. Our approach is based on the general L p Busemann-Petty centroid inequality and does not rely on the general L p Petty projection inequality or the solution of the L p Minkowski problem. A Brothers-Ziemer-type result for the general affine Pólya-Szegö principle is also established. As applications, we reprove some sharp affine Sobolev-type inequalities and settle their equality conditions. We also prove a stability estimate for the affine Sobolev inequality on functions of bounded variation by using our new approach. As a corollary of this stability result, we deduce a stability estimate for the affine logarithmic-Sobolev inequality.

Sharp affine Sobolev type inequalities via the L Busemann–Petty centroid inequality

“…In order to show the efficiency of our approach, we choose three prototypes of sharp affine functional inequalities, namely sharp affine L p log-Sobolev, Sobolev and Gagliardo-Nirenberg inequalities. Our method extends to other situations as can be quoted in the recent works [36], where Nguyen obtains a Pólya-Szegö type principle by using some central ideas of this work, and [12], where the authors introduce and prove the sharp affine L 2 Sobolev trace inequality.…”

Affine vs. Euclidean isoperimetric inequalities

“…The optimal constants c n,p in (2.6) were explicitly computed in [41]. It was later shown by Wang [63] (see also [48] for a different approach) that under mild additional technical assumptions on f , equality holds in (2.6) if and only if f coincides up to translation a.e. on R n with its convex symmetrization f E with respect to an origin-symmetric ellipsoid E in R n (cf.…”

“…on R n with its convex symmetrization f E with respect to an origin-symmetric ellipsoid E in R n (cf. [63] or [48] for definitions). We establish an L p version of Theorem 3 which shows that also the classical and the affine L p Sobolev inequalities are members of a larger family of analytic inequalities parametrized by even, zonal measures on S n−1 .…”

“…If µ is discrete, then inequality (2.7) reduces to (2.6) and if µ is proportional to spherical Lebesgue measure, then (2.7) becomes the classical L p Sobolev inequality of Aubin and Talenti. Using either the approach from [63] or that of [48], it is possible to show that if µ is not discrete, then equality holds in (2.7) if and only if f coincides up to translation a.e. on R n with its symmetric rearrangement f ⋆ .…”

Sharp Geometric Inequalities for the General p-Affine Capacity

“…It is worth to mention that to deal with H p,τ (f ) is much more challenging than H p (f ), mainly because the L p convexifications of level sets of a smooth function f in the latter case always contain the origin in their interiors but in the former may not contain the origin in their interiors. These asymmetric extensions have also been widely used to study affine Sobolev type inequalities, the affine Pólya-Szegö principle as well as many other affine isoperimetric inequalities, see e.g., [31,32,37,39]. In Section 3, we provide several equivalent definitions for the general p-affine capacity, which will be denoted by C p,τ (·).…”