Related papers: A Minkowski problem for $\alpha$-concave functions…
A complete classification of continuous, dually epi-translation invariant, and rotation equivariant valuations on convex functions is established. This characterizes the recently introduced functional Minkowski vectors, which naturally…
This paper describes the theory of Minkowski problems for geometric measures in convex geometric analysis. The theory goes back to Minkowski and Aleksandrov and has been developed extensively in recent years. The paper surveys classical and…
In this paper, we extend the scope of Caffarelli's contraction theorem, which provides a measure of the Lipschitz constant for optimal transport maps between log-concave probability densities in $\R^d$. Our focus is on a broader category of…
In his book on Convex Polyhedra (section 7.2), A.D. Aleksandrov raised a general question of finding variational statements and proofs of existence of polytopes with given geometric data. The first goal of this paper is to give a…
We prove a a formula for the first variation of the integral of a log-concave function, which allows us to define the surface area measure of such a function. The formula holds in complete generality with no regularity assumptions, and is…
We consider the simultaneous optimal transportation of measures, where the target marginal is not necessarily fixed. For this problem, we prove the existence of a solution for completely regular spaces and investigate the structure of the…
We suggest a new way of defining optimal transport of positive-semidefinite matrix-valued measures. It is inspired by a recent rendering of the incompressible Euler equations and related conservative systems as concave maximization…
In Euclidean space, the generalised Minkowski problem asks, for a given finite Radon measure $\mu$ on the unit sphere $\mathbb{S}^d$, to find a compact convex set $K$ with area measure $\mu$. For convex sets in the Minkowski space invariant…
The anisotropic $s$-fractional area measures are introduced as the first variation of the anisotropic fractional $s$-perimeter $P_s(K,L)$, with $L$ an origin symmetric convex body and $s\in(0,1)$. As $s\rightarrow 1^-$, the anisotropic…
The r-parallel set to a set A in a Euclidean space consists of all points with distance at most r from A. We clarify the relation between the volume and the surface area of parallel sets and study the asymptotic behaviour of both quantities…
In contemporary convex geometry, the rapidly developing L_p-Brunn Minkowski theory is a modern analogue of the classical Brunn Minkowski theory. A cornerstone of this theory is the L_p-affine surface area for convex bodies. Here, we…
For a given $1$-Lipschitz map $u\colon\mathbb{R}^n\to\mathbb{R}^m$ we define a partition, up to a set of Lebesgue measure zero, of $\mathbb{R}^n$ into maximal closed convex sets such that restriction of $u$ is an isometry on these sets. We…
We introduce a formulation of optimal transport problem for distributions on function spaces, where the stochastic map between functional domains can be partially represented in terms of an (infinite-dimensional) Hilbert-Schmidt operator…
In this paper, we introduce a Robin boundary analogue of the Orlicz-Minkowski problem, which seeks to find a capillary convex body with a prescribed capillary Orlicz surface area measure in the upper Euclidean half-space. We obtain the…
Minkowski's 2nd theorem in the Geometry of Numbers provides optimal upper and lower bounds for the volume of a $o$-symmetric convex body in terms of its successive minima. In this paper we study extensions of this theorem from two different…
Optimal mass transport, also known as the earth mover's problem, is an optimization problem with important applications in various disciplines, including economics, probability theory, fluid dynamics, cosmology and geophysics to cite a few.…
The Gaussian surface area measures for $C$-pseudo-cones are studied in this paper. Using the variational arguments and the approximation methods of Schneider, we obtain the existence of solutions to the Gaussian-Minkowski problem for…
The concept of angle, angle functions, and the question how to measure angles present old and well-established mathematical topics referring to Euclidean space, and there exist also various extensions to non-Euclidean spaces of different…
If the cost function is not too far from the Euclidean cost, then the optimal map transporting Gaussians restricted to a ball will be regular. \ Similarly, given any cost function which is smooth in a neighborhood of two points on a…
We investigate the problem of pairwise multi-marginal optimal transport, that is, given a collection of probability distributions $\{P_\alpha\}$ on a Polish space $\mathcal{X}$, to find a coupling $\{X_\alpha\}$, $X_\alpha\sim P_\alpha$,…