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 (5)

• General affine surface areas
• A probabilistic approach to the geometry of the \ell_p^n-ball
• Relative entropy of cone measures and $L_p$ centroid bodies
• General Lp affine isoperimetric inequalities
• An Asymmetric Affine Pólya--Szegö Principle

Cited by in corpus (77)

• 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
• On the uniqueness of $L_p$-Minkowski problems: the constant $p$-curvature case in ${\mathbb{R}}^3$
• The $L_p$ dual Minkowski problem for $p>1$ and $q>0$
• The $L_p$-Minkowski problem for $-n<p<1$
• A unified flow approach to smooth, even $L_p$-Minkowski problems
• New Orlicz Affine Isoperimetric Inequalities
• Chord Measures in Integral Geometry and Their Minkowski Problems
• More on logarithmic sums of convex bodies
• Local $L^p$-Brunn-Minkowski inequalities for $p<1$
• Smoothness in the $l_p$ Minkowski problem for $p<1$
• A flow approach to the Musielak-Orlicz-Gauss image problem
• The dual Minkowski problem for symmetric convex bodies
• The log-Brunn-Minkowski inequality in ${\mathbb{R}}^3$
• Orlicz-Minkowski flows
• The Orlicz version of the $L_p$ Minkowski problem on $S^{n-1}$ for $-n<p<0$
• Weighted Minkowski's Existence Theorem and Projection Bodies
• On the $L_p$ Brunn-Minkowski theory and the $L_p$ Minkowski problem for $C$-coconvex sets
• Flow by Gauss curvature to the Aleksandrov and dual Minkowski problems
• Intrinsic and Dual Volume Deviations of Convex Bodies and Polytopes
• On the uniqueness of solutions to the isotropic $L_p$ dual Minkowski problem
• Parabolic approaches to curvature equations
• Variational worn stones
• Wulff shapes and a characterization of simplices via a Bezout type inequality
• Deforming a convex hypersurface by anisotropic curvature flows
• $\mathrm{SL}(\mathrm{n})$ covariant function-valued valuations
• The discrete logarithmic Minkowski problem
• $L^p$-Minkowski Problem under Curvature Pinching
• Equivalence of the local and global versions of the $L^p$-Brunn-Minkowski inequality
• $L_p$-Brunn-Minkowski inequality for $p\in (1-\frac{c}{n^{\frac{3}{2}}},1)$
• Stability of the log-Brunn-Minkowski inequality in the case of many hyperplane symmetries
• The Lp Minkowski problem for polytopes for negative p
• Iterations of curvature images
• Non-uniqueness of solutions to the dual $L_p$-Minkowski problem
• A generalized Gauss curvature flow related to the Orlicz-Minkowski problem
• The (Self-Similar, Variational) Rolling Stones
• Curvature bound for $L_p$ Minkowski problem
• On $L_p$-Brunn-Minkowski type and $L_p$-isoperimetric type inequalities for general measures
• Flow by Gauss curvature to Dual Orlicz-Minkowski problems
• Subspace concentration of dual curvature measures
• A flow approach to the $L_{-2}$ Minkowski problem
• Affine Subspace Concentration Conditions
• The $p$-capacitary Orlicz-Hadamard variational formula and Orlicz-Minkowski problems
• 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
• $C^{1,1}$ regularity for solutions to the degenerate $L_p$ Dual Minkowski problem
• Noncompact $L_p$-Minkowski problems
• On a generalised Blaschke-Santalò inequality
• The $L_p$ Aleksandrov problem for origin-symmetric polytopes
• On the general dual Orlicz-Minkowski problem
• Affine function valued valuations
• The Lp Minkowski problem for polytopes for 0 < p < 1
• On the stability of Brunn-Minkowski type inequalities
• Moments and Valuations
• The $L_p$-Gaussian Minkowski problem
• Orlicz-Legendre Ellipsoids
• Dual curvature measures on convex functions and associated Minkowski problems
• The discrete logarithmic Minkowski problem for the electrostatic $\mathfrak{p}$-capacity
• Stable solution of the log-Minkowski problem in the case of many hyperplane symmetries
• The general dual-polar Orlicz-Minkowski problem
• On the equivalence between two problems of asymmetry on convex bodies
• A flow method for the dual Orlicz-Minkowski problem
• The cone volume measure of antipodal points
• $L^p$ Christoffel-Minkowski problem: the case $1<p<k+1$
• Constrained convex bodies with extremal affine surface areas
• The Minkowski problem based on the (p,q)-mixed quermassintegrals
• A Steiner formula in the $L_p$ Brunn Minkowski theory
• The Minkowski problem in the Gaussian probability space
• The planar $L_p$-Minkowski problem for $0<p<1$
• A note on Bourgain-Milman's universal constant