Related papers: The Logarithmic Minkowski Problem
Existence of solution of the logarithmic Minkowski problem is proved for the case where the discrete measures on the unit sphere satisfy the subspace concentration condition with respect to some special proper subspaces. In order to…
Minkowski's classical existence theorem provides necessary and sufficient conditions for a Borel measure on the unit sphere of Euclidean space to be the surface area measure of a convex body. The solution is unique up to a translation. We…
The Minkowski problem for electrostatic capacity characterizes measures generated by electrostatic capacity, which is a well-known variant of the Minkowski problem. This problem has been generalized to $L_p$ Minkowski problem for…
For a broad class of integral functionals defined on the space of $n$-dimensional convex bodies, we establish necessary and sufficient conditions for monotonicity, and necessary conditions for the validity of a Brunn-Minkowski type…
A variational formula is derived by combining the Gaussian volume of the epigraph of a convex function $\varphi$ and the perturbation of $\varphi$ via the infimal convolution. This formula naturally leads to a Borel measure on…
The dual Minkowski problem for even data asks what are the necessary and sufficient conditions on an even prescribed measure on the unit sphere for it to be the $q$-th dual curvature measure of an origin-symmetric convex body in…
The current state of art concerning the $L_p$ Minkowski problem as a Monge-Ampere equation on the sphere and Lutwak's Logarithmic Minkowski conjecture about the uniqueness of even solution in the $p=0$ case are surveyed and connections to…
The Minkowski problem in convex geometry concerns showing that a given Borel measure on the unit sphere is, up to perhaps a constant, some type of surface area measure of a convex body. Two types of Minkowski problems in particular are an…
The paper characterizes the convex hull of the closure of the cone-volume set $C_\cv(U)$, consisting of all cone-volume vectors of polygons with outer unit normals vectors contained in $U$, for any finite set $U \subseteq \R^2, \pos(U) =…
The notion of ball convexity, considered in finite dimensional real Banach spaces, is a natural and useful extension of usual convexity; one replaces intersections of half-spaces by suitable intersections of balls. A subset $S$ of a normed…
Under study are some vector optimization problems over the space of Minkowski balls, i.e., symmetric convex compact subsets in Euclidean space. A typical problem requires to achieve the best result in the presence of conflicting goals;…
We survey elementary results in Minkowski spaces (i.e. finite dimensional Banach spaces) that deserve to be collected together, and give simple proofs for some of them. We place special emphasis on planar results. Many of these results have…
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,…
Minkowski's Theorem asserts that every centered measure on the sphere which is not concentrated on a great subsphere is the surface area measure of some convex body, and, moreover, the surface area measure determines a convex body uniquely.…
We address an old open question in convex geometry that dates back to the work of Minkowski: what are the equality cases of the monotonicity of mixed volumes? The problem is equivalent to that of providing a geometric characterization of…
The mixed Christoffel-Minkowski problem asks for necessary and sufficient conditions for a Borel measure on the Euclidean unit sphere to be the mixed area measure of some convex bodies, one of which, appearing multiple times, is free and…
In this paper, we establish a necessary condition for the logarithmic Minkowski problem in higher dimensions. This result generalizes a necessary condition proposed by Liu, Lu, Sun, and Xiong in their investigation of the two-dimensional…
A subset of a metric space is a k-distance set if there are exactly k non-zero distances occuring between points. We conjecture that a k-distance set in a d-dimensional Banach space (or Minkowski space), contains at most (k+1)^d points,…
Quantum theory and relativity are the pillar theories on which our understanding of physics is based. Poincar\'e invariance is a fundamental physical principle stating that the experimental results must be the same in all inertial reference…
We prove that an approximated version of the Brunn--Minkowski inequality with volume distortion coefficient implies a Gaussian concentration-of-measure phenomenon. Our main theorem is applicable to discrete spaces.