Geometric measures in the dual Brunn–Minkowski theory and their associated Minkowski problems

2016/01/01 by Yong Huang, Erwin Lutwak, Deane Yang +1 · 255 citations
Mathematics · #Convex body #Convex hull #Dual (grammatical number) #Geometric Analysis and Curvature Flows #Geometry #Mathematical analysis #Mathematics #Minkowski space #Morphological variations and asymmetry #Point processes and geometric inequalities #Pure mathematics #Regular polygon

paper · pdf · doi:10.1007/s11511-016-0140-6

published in Acta Mathematica 216(2), 325-388 (Mittag-Leffler Institute)

openalex publication_date 2016/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A longstanding question in the dual Brunn–Minkowski theory is “What are the dual analogues of Federer’s curvature measures for convex bodies?” The answer to this is provided. This leads naturally to dual versions of Minkowski-type problems: What are necessary and sufficient conditions for a Borel measure to be a dual curvature measure of a convex body? Sufficient conditions, involving measure concentration, are established for the existence of solutions to these problems.

Cited by

• Hadamard variational formulas for p-torsion and p-eigenvalue with applications
• A Gauss curvature flow related to the Orlicz-Aleksandrov problem
• Non-uniqueness of solutions to the dual L_p-Minkowski problem
• Ball-convex bodies and L_p relative surface areas
• The p-th dual Minkowski problem for the k-torsional rigidity corresponding to a k-Hessian equation
• Anisotropic fractional area measures
• Dual curvature measures on convex functions and associated Minkowski problems
• The L_p dual Minkowski problem for capillary hypersurfaces
• The L_p chord Minkowski problem for super-critical exponent
• On The Minkowski Problem With Respect To A Mixed Euclidean-Gaussian Density
• The Minkowski problem based on the (p,q)-mixed quermassintegrals
• The capillary Orlicz-Minkowski problem
• Nonuniqueness and nonexistence results for the Lp-dual Minkowski problem with supercritical exponents
• A flow method for the dual Orlicz-Minkowski problem
• The general dual-polar Orlicz-Minkowski problem
• On the Musielak-Orlicz-Gauss image problem
• Long time behavior of a class of non-homogeneous anisotropic fully nonlinear curvature flows
• On a general class of free boundary Monge-Ampère equations
• Flow by Gauss curvature to Dual Orlicz-Minkowski problems
• The L_p Dual Minkowski Problem for Group-Invariant Convex Bodies
• On a generalised Blaschke-Santalò inequality
• The L_p Aleksandrov problem for origin-symmetric polytopes
• A Minkowski problem for α-concave functions via optimal transport
• Subspace concentration of dual curvature measures
• Brunn-Minkowski and Reverse Isoperimetric Inequalities for Dual Quermassintegrals
• The dual Orlicz-Minkowski problem
• The Minkowski problem in the Gaussian probability space
• Constrained convex bodies with extremal affine surface areas
• The uniqueness and non-uniqueness of solutions to the even dual Minkowski problem
• On the general dual Orlicz-Minkowski problem
• Compactness of the L_p dual Minkowski problem in ℝ³
• Capillary curvature images
• SL(n) contravariant tensor valuations of small orders
• A Minkowski Theory for the Exterior Capacitary Volumes and A Resolution of the Pólya-Szegö Conjecture
• The capillary L_p-Minkowski problem
• Uniqueness in the near isotropic Lp dual Minkowski problem
• The L_p-Gaussian Minkowski problem
• Minkowski valuations on convex functions
• Parabolic approaches to curvature equations
• The L_p Minkowski problems on affine dual quermassintegrals
• Empirical dual volumes
• Intrinsic and Dual Volume Deviations of Convex Bodies and Polytopes
• Deforming a Convex Hypersurface by Anisotropic Curvature Flows
• The Vectorial Hadwiger Theorem on Convex Functions
• The L_p dual Christoffel-Minkowski problem for 1
• Minkowski valuations on convex functions. [europepmc]
• Some inequalities for ( p , q ) -mixed volume. [europepmc]
• The Hadwiger theorem on convex functions, III: Steiner formulas and mixed Monge-Ampère measures. [europepmc]

Related

• Minkowski Problems for Geometric Measures
• Dual curvature measures on convex functions and associated Minkowski problems
• Uniqueness and continuity of the solution to L_p dual Minkowski problem
• The L_p dual Minkowski problem for p>1 and q>0
• The dual Minkowski problem for symmetric convex bodies