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

Abstract

We show that the Lp Busemann-Petty centroid inequality provides an elementary and powerful tool to the study of some sharp affine functional inequalities with a geometric content, like log-Sobolev, Sobolev and Gagliardo-Nirenberg inequalities. Our approach allows also to characterize directly the corresponding equality cases.2010 Mathematics Subject Classification. Primary 46E35, Secondary 51M16.

Asymmetric Blaschke-Santaló functional inequalities

“…The family of asymmetric centroid inequalities (8) was proved by Haberl and Schuster in [7]. Notice that h(Mε,pK, ·) as given in the first formula is always a convex function regardless of K ⊂ R n being convex or not.…”

“…Since M • ε,p f is a convex body and 0 is the Santaló point of Mε,pM • ε,p f we may apply Inequality (8)…”

“…denotes the best Sobolev constant for (12). Following [8], if we choose a specific norm . f for each f and apply the Lp Busemann-Petty centroid inequality, we end up with the important sharp affine Sobolev inequality of Lutwak, Yang and Zhang [14] f np…”

Sharp affine weighted 𝐿^{𝑝} Sobolev type inequalities

“…Inequality (4) has been proved by using an alternative method introduced recently by Haddad, Jiménez and Montenegro [24] which uses the L p Busemann-Petty centroid inequality as a fundamental tool and has the advantage of not depending on the solution to the L p Minkowski problem. The efficiency of method is also illustrated in [24] with an alternative proof of the well known sharp affine L p Gagliardo-Nirenberg and log-Sobolev inequalities.…”

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

“…In particular, on the L p Petty projection inequality [44] (see [9] for an alternative proof) and on the solution of the normalized L p Minkowski problem [46].The main aim of the present paper is to give a new proof for these affine Pólya-Szegötype principles. Our approach is based on a recent work of Haddad, Jiménez and Montenegro (see [34]) where they give a new proof of some sharp (symmetric) affine Sobolev-type inequalities (like Sobolev, Gagliardo-Nirenberg and logarithmic-Sobolev inequalities [45,64]) by using the L p Busemann-Petty centroid inequality [44]. We show that their method also can be applied to general cases considered in [32,33,62], and hence gives an alternative proof for the results in [13,33,62].…”