Busemann-Petty Centroid Inequality Research Articles

$$L_p$$ Busemann–Petty centroid inequality in hyperbolic and spherical spaces

$$L_p$$ Busemann–Petty centroid inequality in hyperbolic and spherical spaces

Lp centroid bodies with respect to weights |x|ℓ

The Lp (where 1≤p≤∞) centroid bodies with respect to weights that are powers of the distance to the origin (i.e., |x|ℓ with ℓ>−n) and their associated isoperimetric inequality (i.e., the polar Lp Busemann-Petty centroid inequality) are introduced in this paper. By means of Ball's body, a corresponding functional version of the polar Lp Busemann-Petty centroid inequality will be also studied.

Sharp affine Trudinger–Moser inequalities: A new argument

AbstractWe set up the sharp Trudinger–Moser inequality under arbitrary norms. Using this result and the $L_{p}$ Busemann-Petty centroid inequality, we will provide a new proof to the sharp affine Trudinger–Moser inequalities without using the well-known affine Pólya–Szegö inequality.

An L p-Functional Busemann–Petty Centroid Inequality

Abstract For a convex body $K\subset \mathbb{R}^n$, let $\Gamma _pK$ be its $L_p$-centroid body. The $L_p$-Busemann–Petty centroid inequality states that $\operatorname{vol}(\Gamma _pK) \geq \operatorname{vol}(K)$, with equality if and only if $K$ is an ellipsoid centered at the origin. In this work, we prove inequalities for a type of functional $L_r$-mixed volume for $1 \leq r < n$ and establish, as a consequence, a functional version of the $L_p$-Busemann–Petty centroid inequality.

Complex affine isoperimetric inequalities

Complex extensions of the Petty projection inequality and the Busemann–Petty centroid inequality are established.

On the reverse Orlicz–Lorentz Busemann–Petty centroid inequality

This paper studies the extrema of some affine invariant functionals related to the volume of the Orlicz–Lorentz centroid body introduced by Nguyen. We obtain some variants of the Orlicz–Lorentz Busemann–Petty centroid inequality, and also prove the reverse form of these inequalities in the two-dimensional case.

The sharp affine $$L^2$$ L 2 Sobolev trace inequality and variants

We establish a sharp affine $$L^p$$ Sobolev trace inequality by using the $$L_p$$ Busemann–Petty centroid inequality. For $$p = 2$$ , our affine version is stronger than the famous sharp $$L^2$$ Sobolev trace inequality proved independently by Escobar and Beckner. Our approach allows also to characterize all extremizers in this case. For this new inequality, no Euclidean geometric structure is needed.

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

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 Lp Busemann–Petty centroid inequality and does not rely on the general Lp Petty projection inequality or the solution of the Lp 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 Lp Busemann–Petty centroid inequality

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.

The planar Busemann-Petty centroid inequality and its stability

In Centro-affine invariants for smooth convex bodies [Int. Math. Res. Notices. DOI 10.1093/imrn/rnr110, 2012] Stancu introduced a family of centro-affine normal flows, $p$-flow, for $1\leq p<\infty .$ Here we investigate the asymptotic behavior of the planar $p$-flow for $p=\infty$, in the class of smooth, origin-symmetric convex bodies. First, we prove that the $\infty$-flow evolves appropriately normalized origin-symmetric solutions to the unit disk in the Hausdorff metric, modulo $SL(2).$ Second, using the $\infty$-flow and a Harnack estimate for this flow, we prove a stability version of the planar Busemann-Petty centroid inequality in the Banach-Mazur distance. Third, we prove that the convergence of normalized solutions in the Hausdorff metric can be improved to convergence in the $\mathcal {C}^{\infty }$ topology.

On the reverse Orlicz Busemann–Petty centroid inequality

In this paper, the Orlicz centroid body, defined by E. Lutwak, D. Yang and G. Zhang, and the extrema of some affine invariant functionals involving the volume of the Orlicz centroid body are investigated. The reverse form of the Orlicz Busemann–Petty centroid inequalities is obtained in the two-dimensional case.

A new proof of the Orlicz Busemann-Petty centroid inequality

Using shadow systems, we provide a new proof of the Orlicz Busemann-Petty centroid inequality, which was first obtained by Lutwak, Yang and Zhang.

On reverses of the L p -Busemann-Petty centroid inequality and its applications

In this paper, the reverse forms of the Lp-Busemann-Petty centroid inequality are shown. As the applications of the reverse forms, we obtain the reverse forms of the Lp-centroid-affine inequality and an upper bound of the isotropic constant for convex bodies.

Orlicz centroid bodies

The sharp affine isoperimetric inequality that bounds the volume of the centroid body of a star body (from below) by the volume of the star body itself is the Busemann-Petty centroid inequality. A decade ago, the $L_p$ analogue of the classical Busemann- Petty centroid inequality was proved. Here, the definition of the centroid body is extended to an Orlicz centroid body of a star body, and the corresponding analogue of the Busemann-Petty centroid inequality is established for convex bodies.

General $L_p$ affine isoperimetric inequalities

Sharp $L_p$ affine isoperimetric inequalities are established for the entire class of $L_p$ projection bodies and the entire class of $L_p$ centroid bodies. These new inequalities strengthen the $L_p$ Petty projection and the $L_p$ Busemann–Petty centroid inequality.

Equivalence of Some Affine Isoperimetric Inequalities

We establish the equivalence of some affine isoperimetric inequalities which include the -Petty projection inequality, the -Busemann-Petty centroid inequality, the dual -Petty projection inequality, and the dual -Busemann-Petty inequality. We also establish the equivalence of an affine isoperimetric inequality and its inclusion version for -John ellipsoids.

Lp Affine Isoperimetric Inequalities

The Lp analogues of the Petty projection inequality and the BusemannPetty centroid inequality are established. An affine isoperimetric inequality compares two functionals associated with convex (or more general) bodies, where the ratio of the functionals is invariant under non-degenerate linear transformations. These isoperimetric inequalities are more powerful than their better-known Euclidean relatives.

Mixed projection inequalities

A number of sharp geometric inequalities for polars of mixed projection bodies (zonoids) are obtained. Among the inequalities derived is a polar projection inequality that has the projection inequality of Petty as a special case. Other special cases of this polar projection inequality are inequalities (between the volume of a convex body and that of the polar of its i i th projection body) that are strengthened forms of the classical inequalities between the volume of a convex body and its projection measures (Quermassintegrale). The relation between the Busemann-Petty centroid inequality and the Petty projection inequality is shown to be similar to the relation that exists between the Blaschke-Santaló inequality and the affine isoperimetric inequality of affine differential geometry. Some mixed integral inequalities are derived similar in spirit to inequalities obtained by Chakerian and others.