The dual L_p John ellipsoid, including the classical Löwner and Legendre ellipsoids, arises as the solution to a certain optimization problem. In this paper, we consider a broad extension within the framework of the dual weighted L_p Brunn-Minkowski theory. A variational formula for general measures with locally integrable densities, under L_p-harmonic radial combinations, is established. This leads us to propose a corresponding optimization problem for general measures. We prove the existence, uniqueness and characterization of the solution to this problem, which defines a new type of ellipsoid. This ellipsoid extends the dual L_p John ellipsoid to a significantly general more setting, including Gaussian measures. The related geometric inequalities and the L_p Löwner inclusion for general measures are also given.

In this paper, we establish the functional forms of the dual Blaschke-Santaló inequality and its inverse, which are the dual counterparts for the classical functional Blaschke-Santaló inequality and its inverse.

In this article, we reformulate LXYZ's $Lp$ affine Sobolev inequality chain (including Lutwak-Yang-Zhang's $Lp$ affine Sobolev inequality and Xiao's p-affine capacity inequality) in the setting of Grassmann manifolds. For this purpose, the Grassmannian $ p $ -affine capacity is introduced.

In this article, we reformulate LXYZ’s $L^p$ affine Sobolev inequality chain (including Lutwak–Yang–Zhang’s $L^p$ affine Sobolev inequality and Xiao’s p -affine capacity inequality) in the setting of Grassmann manifolds. For this purpose, the Grassmannian $ p $ -affine capacity is introduced.

Based on the mixed cone-volume measure, we introduce in this paper a new ellipsoid, called the mixed-sine Lutwak-Yang-Zhang (LYZ) ellipsoid, which serves as a sine counterpart to the mixed LYZ ellipsoid. Several related volume inequalities for this ellipsoid are also established.

In this paper, we establish stability versions of the volume inequalities associated with Lp zonoids. These results, particularly for the case 1 <= p <= 2, extend the case p=1 previously obtained by Brazitikos and Giannopoulos. As applications, we derive several stability inequalities for Lp isotropic convex bodies and for bodies with minimal p-mean width and minimal Lp surface area.

The main purpose of this paper is to establish the Blaschke-Santal & oacute; inequality for the normalized Lp Busemann body. The approach we take is from the functional point of view. By applying the functional Lp-Busemann random simplex inequality due to Dann et al. (2016), we establish the Blaschke-Santal & oacute; inequality for the functional Lp Busemann body and corresponding moment-entropy inequalities. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

In this paper, we establish the functional sine Blaschke-Santalo inequality, which can be viewed as the "sine cousin" of the celebrated functional Blaschke-Santalo inequality.

In this paper, we establish a sine version of the moment-entropy inequality due to Lutwak et al. [8]. Moreover, we show that this inequality is equivalent to the L-p-sine moment-entropy inequality recently established in [6].

In this paper, we explore some properties of the dual Wills functional, which are part of the dual Brunn–Minkowski theory. We give the upper and lower bounds for the dual Wills functional in terms of the 1-th dual volume of star bodies. Moreover, an inequality that is associated with the section of convex bodies for isotropic measures is presented.

Based on the mixed Grassmannian L_p -Busemann random simplex inequality, the Grassmannian L_p -sine Blaschke–Santaló inequality is established in this paper. This extends the recently obtained L_p -sine Blaschke–Santaló inequality to the Grassmannian form.

In this paper,we establish volume inequalities for k-dimensional sections and projections of convex bodies(not necessarily symmetric)and their polars in a more general position than John's position.

运用Brascamp-Lieb不等式的几何版本及其逆不等式,建立了中心对称凸体的k-维截面及其极体投影的体积不等式,其位置比John位置更一般.

As one application of the Cauchy–Binet formula, the Ball–Barthe inequality plays a key role in solving reverse isoperimetric inequalities. Recently, a new Grassmannian Ball–Barthe inequality was established. With it, reverse isoperimetric inequalities on Grassmann manifolds were successfully proved. In this paper, we will establish its matrix form with a simpler proof than that of Grassmannian form. The matrix form also recovers the classical Ball–Barthe inequality in a special case.

In this paper, we show that the volume of a k-dimensional ellipsoid in the convex body formed by centered isotropic measures on the unit sphere is no large than that of a k-dimensional Ball of radius root n(n + 1)/k(k + 1). It generalizes the John theorem to the lower dimensional cases.

This paper is dedicated to study the sine version of polar bodies and establish the L-p-sine Blaschke-Santalo inequality for the L-p-sine centroid body. The L-p-sine centroid body Lambda K-p for a star body K subset of R-n is a convex body based on the L-p-sine transform, and its associated Blaschke-Santalo inequality provides an upper bound for the volume of Lambda(p)degrees K, the polar body of Lambda K-p, in terms of the volume of K. Thus, this inequality can be viewed as the "sine cousin" of the L-p Blaschke-Santalo inequality established by Lutwak and Zhang. As p -> infinity, the limit of Lambda(p)degrees K becomes the sine polar body K-lozenge and hence the L-p-sine Blaschke-Santalo inequality reduces to the sine Blaschke-Santalo inequality for the sine polar body. The sine polarity naturally leads to a new class of convex bodies Cen, which consists of all originsymmetric convex bodies generated by the intersection of origin-symmetric closed solid cylinders. Many notions in C-e(n) are developed, including the cylindrical support function, the supporting cylinder, the cylindrical Gauss image, and the cylindrical hull. Based on these newly introduced notions, the equality conditions of the sine Blaschke-Santalo inequality are settled. (c) 2022 Elsevier Inc. All rights reserved.

In this paper, we establish mean width inequalities of sections and projections of convex bodies for isotropic measures with complete equality conditions, which extends the recent work of Alonso-Gutiérrez and Brazitikos. Different from their approach, our proof is based on the approach developed by Lutwak, Yang and Zhang, by using the Ball-Barthe inequality, the mass transportation, and the isotropic embedding.

In this paper, we establish mean width inequalities of sections and projections of convex bodies for isotropic measures with complete equality conditions, which extends the recent work of Alonso-Guti\'{e}rrez and Brazitikos. Different from their approach, our proof is based on the approach developed by Lutwak, Yang and Zhang, by using the Ball-Barthe inequality, the mass transportation, and the isotropic embedding.

Based on reverse isoperimetric inequalities on Grassmann manifolds, a Grassmanian Loomis-Whitney inequality and its dual are established, which provides a lower bound for the volume of an origin-symmetric convex body in terms of its lower dimensional sections.