On the mth order p-affine capacity

2025/05/18 by Xiaoyan Zhou, Zhou, Xia, Deping Ye +1
Mathematics · Computer Science · #Point processes and geometric inequalities #Geometric Analysis and Curvature Flows #Optimization and Variational Analysis

paper · pdf · doi:10.48550/arxiv.2505.12573

Abstract

Let M_{n, m}(ℝ) denote the space of n× m real matrices, and K_o^n,m be the set of convex bodies in M_{n, m}(ℝ) containing the origin. We develop a theory for the mth order p-affine capacity C_p,Q(⋅) for p∈[1,n) and Q\inK_o^1,m. Several equivalent definitions for the mth order p-affine capacity will be provided, and some of its fundamental properties will be proved, including for example, translation invariance and affine invariance. We also establish several inequalities related to the mth order p-affine capacity, including those comparing to the p-variational capacity, the volume, the mth order p-integral affine surface area, as well as the L_p surface area.

Citations

• Some comments on the mth-order Projection Bodies
• On the polar of Schneider's difference body
• On general versions of the Petty projection inequality
• On Moment-Entropy inequalities in the space of matrices
• The mth order Orlicz projection bodies
• The L_p-Brunn-Minkowski inequalities for variational functionals with 0≤ p<1
• On the mth-order Affine Pólya-Szegö Principle
• Higher-Order Reverse Isoperimetric Inequalities for Log-concave Functions
• General Higher Order L^p Mean Zonoids
• A Rogers--Brascamp--Lieb--Luttinger inequality in the space of matrices
• Affine Isoperimetric Inequalities for Higher-Order Projection and Centroid Bodies
• Affine Fractional Sobolev and Isoperimetric Inequalities
• On the discrete functional l_p Minkowski problem
• Sharp geometric inequalities for the general p-affine capacity
• The p-capacitary Orlicz-Hadamard variational formula and Orlicz-Minkowski problems
• A unified treatment for Lp Brunn-Minkowski type inequalities
• An Asymmetric Affine Pólya--Szegö Principle
• General Lp affine isoperimetric inequalities
• Normal and Integral Currents

Related

• Sharp geometric inequalities for the general p-affine capacity
• Constrained convex bodies with extremal affine surface areas
• Convex bodies and algebraic equations on affine varieties
• p-capacity vs surface-area
• Sharp upper bounds for the capacity in the hyperbolic and Euclidean spaces