An extension of Minkowski's theorem and its applications to questions about projections for measures

DOI10.1016/j.aim.2019.106803zbMath1429.52004arXiv1607.06531OpenAlexW2974206593MaRDI QIDQ2326664

Galyna V. Livshyts

Publication date: 10 October 2019

Published in: Advances in Mathematics (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/1607.06531

zbMATH Keywords

convex bodieslog-concaveMinkowski theoremAlexandrov's problemBrunn-Minkowskicone-measureShepherd's problem

Mathematics Subject Classification ID

Lattices and convex bodies in (n) dimensions (aspects of discrete geometry) (52C07) Convex sets in (n) dimensions (including convex hypersurfaces) (52A20)

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Cites Work

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• The spherical convex floating body
• A hyperplane inequality for measures of convex bodies in \(\mathbb R^{n }, \; n\leq 4\)
• The log-Brunn-Minkowski inequality
• Stability in the Busemann-Petty and Shephard problems
• The Brunn-Minkowski-Firey theory. I: Mixed volumes and the Minkowski problem
• Intersection bodies and dual mixed volumes
• Critères de convexité et inégalités intégrales
• A positive solution to the Busemann-Petty problem in \(\mathbb{R}^4\)
• An analytic solution to the Busemann-Petty problem on sections of convex bodies
• A generalization of the Busemann-Petty problem on sections of convex bodies
• Contributions to affine surface area
• Ellipsoids and matrix-valued valuations
• On the number of solutions to the discrete two-dimensional \(L_{0}\)-Minkowski problem.
• The Ehrhard inequality
• On the \(p\)-affine surface area
• A new ellipsoid associated with convex bodies
• The discrete planar \(L_0\)-Minkowski problem
• \(L_ p\) affine isoperimetric inequalities.
• A discrete version of Koldobsky's slicing inequality
• The Busemann-Petty problem for arbitrary measures
• The Cramer-Rao inequality for star bodies
• Projections of convex bodies and the Fourier transform
• On the regularity of solutions to a generalization of the Minkowski problem
• The Brunn-Minkowski-Firey theory. II: Affine and geominimal surface areas
• Complemented Brunn-Minkowski inequalities and isoperimetry for homogeneous and non-homogeneous measures
• The logarithmic Minkowski problem for polytopes
• An isomorphic version of the Busemann-Petty problem for arbitrary measures
• Convex set functions in \(d\)-space
• Affine images of isotropic measures
• The \(L_p\) Minkowski problem for polytopes for \(0 < p < 1\)
• On the uniqueness of \(L_p\)-Minkowski problems: the constant \(p\)-curvature case in \(\mathbb{R}^3\)
• On the stability of Brunn-Minkowski type inequalities
• On the extension of operators with a finite-dimensional range
• Shadow systems of convex sets
• Zu einem Problem von Shephard über die Projektionen konvexer Körper
• Fourier transforms and the Funk-Hecke theorem in convex geometry
• Symétrisation dans l'espace de Gauss.
• Inequalities between projection functions of convex bodies
• A Representation of the Symmetric Bivariate Cauchy Distribution
• Shadows of Convex Bodies
• Intersection bodies, positive definite distributions, and the Busemann-Petty problem
• The Brunn-Minkowski inequality
• On the $L_{p}$-Minkowski problem
• Fourier transform and Firey projections of convex bodies
• The logarithmic Minkowski problem
• Maximal Surface Area of a Convex Set in $$\mathbb{R}^{n}$$ with Respect to Log Concave Rotation Invariant Measures
• A Class of Negative-Definite Functions