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We introduce dual curvature measures for log-concave functions, which in the case of characteristic functions recover the dual curvature measures for convex bodies introduced by Huang-Lutwak-Yang-Zhang in 2016. Variational formulas are…
This paper presents a brief overview of ravine functions using the example of the Minkowski-Cohn moduli surface from the point of view of optimization on it. Elements of representation and solution of the minimization problem at a point are…
We propose a new approach to deriving quantitative mean field approximations for any probability measure $P$ on $\mathbb{R}^n$ with density proportional to $e^{f(x)}$, for $f$ strongly concave. We bound the mean field approximation for the…
We present a general convex relaxation approach to study a wide class of Unbalanced Optimal Transport problems for finite non-negative measures with possibly different masses. These are obtained as the lower semicontinuous and convex…
We construct and investigate smooth orientable surfaces in su(N) algebras. The structural equations of surfaces associated with Grassmannian sigma models on Minkowski space are studied using moving frames adapted to the surfaces. The first…
We introduce a general class of transport distances ${\rm WB}_{\Lambda}$ over the space of positive semi-definite matrix-valued Radon measures $\mathcal{M}(\Omega,\mathbb{S}_+^n)$, called the weighted Wasserstein-Bures distance. Such a…
The aim of this paper is to study properties of sections of convex bodies with respect to different types of measures. We present a formula connecting the Minkowski functional of a convex symmetric body K with the measure of its sections.…
This is the lecture notes on the interplay between optimal transport and Riemannian geometry. On a Riemannian manifold, the convexity of entropy along optimal transport in the space of probability measures characterizes lower bounds of the…
The stability of solutions to optimal transport problems under variation of the measures is fundamental from a mathematical viewpoint: it is closely related to the convergence of numerical approaches to solve optimal transport problems and…
In this note, we extend the regularity theory for monotone measure-preserving maps, also known as optimal transports for the quadratic cost optimal transport problem, to the case when the support of the target measure is an arbitrary convex…
On some specified convex supporting sets of spheres, we find a generalized longitude function whose level sets are totally geodesic. Given an arbitrary (weakly) harmonic map into spheres, the composition of the generalized longitude…
The Gromov--Wasserstein problem is a non-convex optimization problem over the polytope of transportation plans between two probability measures supported on two spaces, each equipped with a cost function evaluating similarities between…
We study one-loop perturbative properties of scalar field theories on the $\rho$-Minkowski space. The corresponding star-product, together with the involution are characterized from a combination of Weyl quantization and defining properties…
We develop and study a theory of optimal transport for vector measures. We resolve in the negative a conjecture of Klartag, that given a vector measure on Euclidean space with total mass zero, the mass of any transport set is again zero. We…
The Penrose inequality in Minkowski is a geometric inequality relating the total outer null expansion and the area of closed, connected and spacelike codimension-two surfaces S in the Minkowski spacetime, subject to an additional convexity…
The aim of this article is to introduce and address the problem to smoothly interpolate (empirical) probability measures. To this end, we lift the concept of a spline curve from the setting of points in a Euclidean space that that of…
We do further investigation in a certain cosine function defined for smooth Minkowski spaces. We prove that such function is symmetric if and only if the referred space is Euclidean, and also that it can be given in terms of the Gateaux…
This paper addresses the Optimal Transport problem, which is regularized by the square of Euclidean $\ell_2$-norm. It offers theoretical guarantees regarding the iteration complexities of the Sinkhorn--Knopp algorithm, Accelerated Gradient…
In this paper, the Orlicz addition of measures is proposed and an interpretation of the $f$-divergence is provided based on a linear Orlicz addition of two measures. Fundamental inequalities, such as, a dual functional…
The general theory for quantum simulation of cubic semiconductor n-MOSFETs is presented within the effective mass equation approach. The full three-dimensional transport problem is described in terms of coupled transverse subband modes…