The Logarithmic Minkowski Problem

Abstract

In analogy with the classical Minkowski problem, necessary and sufficient conditions are given to assure that a given measure on the unit sphere is the cone-volume measure of the unit ball of a finite dimensional Banach space.

References in corpus (3)

• A probabilistic approach to the geometry of the \ell_p^n-ball
• General Lp affine isoperimetric inequalities
• An Asymmetric Affine Pólya--Szegö Principle

Cited by in corpus (43)

• Geometric measures in the dual Brunn-Minkowski theory and their associated Minkowski problems
• The dual Minkowski problem for negative indices
• The Dual Orlicz-Brunn-Minkowski Theory
• New Orlicz Affine Isoperimetric Inequalities
• Chord Measures in Integral Geometry and Their Minkowski Problems
• A flow approach to the Musielak-Orlicz-Gauss image problem
• The dual Minkowski problem for symmetric convex bodies
• On the $L_p$ Brunn-Minkowski theory and the $L_p$ Minkowski problem for $C$-coconvex sets
• Weighted Minkowski's Existence Theorem and Projection Bodies
• Flow by Gauss curvature to the Aleksandrov and dual Minkowski problems
• On the uniqueness of solutions to the isotropic $L_p$ dual Minkowski problem
• Deforming a convex hypersurface by anisotropic curvature flows
• Variational worn stones
• The discrete logarithmic Minkowski problem
• $\mathrm{SL}(\mathrm{n})$ covariant function-valued valuations
• $L^p$-Minkowski Problem under Curvature Pinching
• Equivalence of the local and global versions of the $L^p$-Brunn-Minkowski inequality
• The (Self-Similar, Variational) Rolling Stones
• A generalized Gauss curvature flow related to the Orlicz-Minkowski problem
• Non-uniqueness of solutions to the dual $L_p$-Minkowski problem
• Curvature bound for $L_p$ Minkowski problem
• Flow by Gauss curvature to Dual Orlicz-Minkowski problems
• Affine Subspace Concentration Conditions
• The Lp Minkowski problem for polytopes for 0 < p < 1
• The dual Orlicz-Minkowski problem
• Lp Minkowski problem for electrostatic $\mathfrak{p}$-capacity
• Geometry of log-concave functions: the $L_p$ Asplund sum and the $L_p$ Minkowski problem
• A non-existence result for the $L_p$-Minkowski problem
• Prescribed $L_p$ curvature problem
• On the Minkowski problem for p-harmonic measures
• The Gauss Image Problem with Weak Aleksandrov Condition
• On the Musielak-Orlicz-Gauss image problem
• A flow method for the dual Orlicz-Minkowski problem
• $L^p$ Christoffel-Minkowski problem: the case $1<p<k+1$
• The planar $L_p$-Minkowski problem for $0<p<1$
• Dual curvature measures on convex functions and associated Minkowski problems
• A note on Bourgain-Milman's universal constant
• The discrete logarithmic Minkowski problem for the electrostatic $\mathfrak{p}$-capacity
• Moments and Valuations
• The cone volume measure of antipodal points
• The $L_p$-Gaussian Minkowski problem
• Stable solution of the log-Minkowski problem in the case of many hyperplane symmetries
• The general dual-polar Orlicz-Minkowski problem