Equivalence of Some Affine Isoperimetric Inequalities

About a decade ago Lutwak, Yang, and Zhang introduced the notion of $L_p$-projection body. More recently, Wang and Leng established an $L_p$-version of Petty's affine projection inequality. At the same time Ludwig discovered a family of general $L_p$-projection bodies and Haberl and Schuster established Petty's projection inequality for general $L_p$-projection bodies. In this paper we establish a general $L_p$-version of Petty's affine projection inequality for general $L_p$-projection bodies. Moreover, we obtain an analogous inequality for $L_p$-geominimal surface area.

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.

Using results of K. Kiener and the Riesz-Sobolev convolution inequality we give a new proof of Petty's projection inequality. By the same method we also obtain a proof of Santalo's affine isoperimetric inequality.

The classical Petty projection inequality is an affine isoperimetric inequality which constitutes a cornerstone in the affine geometry of convex bodies. By extending the polar projection body to an inter‐dimensional operator, Petty's inequality was generalized in Haddad, Langharst, Putterman, Roysdon, and Ye to the so‐called setting, where is an ‐dimensional compact convex set. In this work, we further extend the Petty projection inequality to the broader realm of rotationally invariant measures with concavity properties, namely, those with ‐concave density (for ). Moreover, when , and motivated by a contemporary empirical reinterpretation of Petty's result by Paouris, Pivovarov, and Tatarko, we explore empirical analogues of this inequality.

Affine vs. Euclidean isoperimetric inequalities

Recently, Lutwak, Yang and Zhang posed the notion of L p -projection body and established the L p -analog of the Petty projection inequality. In this paper, the notion of L p -mixed projection body is introduced—the L p -projection body being a special case. The Petty projection inequality, as well as Lutwak's quermassintegrals (L p -mixed quermassintegrals) extension of the Petty projection inequality, is established for L p -mixed projection body.

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

The ‐th dual curvature measure was introduced by Lutwak, Yang and Zhang. In this paper, we study the cosine transform of the ‐th dual curvature measure which defines a new convex body. We prove that the new convex body unifies the Petty projection body and the centroid body. The corresponding affine isoperimetric inequality is also obtained. It is an extension of the known Petty projection inequality.

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.

We extend the study of the functional L p L_p -Busemann-Petty centroid inequality by addressing an endpoint case, complementing the results of Haddad et al. [Int. Math. Res. Not. IMRN 10 (2021), pp. 7947–7965].

Petty's conjectured projection inequality is a famous open problem in convex bodies theory. In this paper, it is shown that a $L_p$-version of the Petty's conjectured projection inequality. As its applications, we give a reverse of the Blaschke-Santaló inequality and consider the monotony of volumes for convex body and its $L_p$-Petty projection body, respectively. Otherwise, we also give the reverses of the $L_p$-Petty projection inequality.

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.

We define a spherical and hyperbolic analog to the Euclidean projection body for star bodies via the gnomonic projection from the unit sphere and stereographic projection in the hyperbolid model of hyperbolic space. We then prove a spherical and hyperbolic projection inequality for these notions by using an adaption of Steiner symmetrization for spherical, respectively, hyperbolic, star bodies.

By using inequalities obtained for the volume of mixed bodies and the Petty Projection Inequality, (sharp) isoperimetric inequalities are derived for the projection measures (Quermassintegrale) of a convex body. These projection measure inequalities, which involve mixed projection bodies (zonoids), are shown to be strengthened versions of the classical inequalities between the projection measures of a convex body. The inequality obtained for the volume of mixed bodies is also used to derive a form of the Brunn-Minkowski inequality involving mixed bodies. As an application, inequalities are given between the projection measures of convex bodies and the mixed projection integrals of the bodies.

By using inequalities obtained for the volume of mixed bodies and the Petty Projection Inequality, (sharp) isoperimetric inequalities are derived for the projection measures (Quermassintegrale) of a convex body, These projection measure inequalities, which involve mixed projection bodies (zonoids), are shown to be strengthened versions of the classical inequalities between the projection measures of a convex body, The inequality obtained for the volume of mixed bodies is also used to derive a form of the Brunn-Minkowski inequality involving mixed bodies, As an application, inequalities are given between the projection measures of convex bodies and the mixed projection integrals of the bodies.