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## Eigenvalue distribution of optimal transportation.

arxiv.org.
math.
Cornell University
,
2013.
No. 1402.2636.

We investigate the Brenier map \nabla \Phi between the uniform measures on two convex domains in \mathbb{R}^n or more generally, between two log-concave probability measures on \mathbb{R}^n. We show that the eigenvalues of the Hessian matrix D^2 \Phi exhibit remarkable concentration properties on a multiplicative scale, regardless of the choice of the two measures or the dimension n.

Calculus of Variations and Partial Differential Equations 2014 Vol. 49 No. 3-4 P. 1187-1197

Let \mu = e^{-V} \ dx be a probability measure and T = \nabla \Phi be the optimal transportation mapping pushing forward \mu onto a log-concave compactly supported measure \nu = e^{-W} \ dx. In this paper, we introduce a new approach to the regularity problem for the corresponding Monge--Amp{\`e}re equation e^{-V} = \det D^2 ...

Added: March 12, 2014

Discrete and Continuous Dynamical Systems 2014 Vol. 34 No. 4 P. 1511-1532

We study the optimal transportation mapping VΦ: ℝd → ℝd pushing forward a probability measure μ = e -V dx onto another probability measure ν = e-W dx. Following a classical approach of E. Calabi we introduce the Riemannian metric g = D2 Φ on ℝd and study spectral properties of the metric-measure space M ...

Added: November 12, 2013

Hessian manifolds, CD(K,N) -spaces, and optimal transportation of log-concave measures / Cornell University. Series math "arxiv.org". 2012. No. 1201.2342.

We study the optimal transportation mapping $\nabla \Phi : \mathbb{R}^d \mapsto \mathbb{R}^d$ pushing forward a probability measure $\mu = e^{-V} \ dx$ onto another probability measure $\nu = e^{-W} \ dx$. Following a classical approach of E. Calabi we introduce the Riemannian metric $g = D^2 \Phi$ on $\mathbb{R}^d$ and study spectral properties of the ...

Added: March 28, 2013

Теория вероятностей и ее применения 2012 Т. 57 № 2 С. 296-321

We study Sobolev a priori estimates for the optimal transportation $T = \nabla \Phi$ between probability measures $\mu=e^{-V} \ dx$ and $\nu=e^{-W} \ dx$ on $\R^d$.
Assuming uniform convexity of the potential $W$ we show that $\int \| D^2 \Phi\|^2_{HS} \ d\mu$, where $\|\cdot\|_{HS}$ is the Hilbert-Schmidt norm,
is controlled by the Fisher information of $\mu$. In ...

Added: February 19, 2013

Pinsker inequalities and related Monge-Amp`ere equations for log-concave functions / Cornell University. Series arXiv "math". 2021.

We further develop the theory of f-divergences for log-concave functions and their related inequalities. We establish Pinsker inequalities, new affine invariant entropy inequalities and new inequalities on functional affine surface area The functional inequalities lead to new affine invariant inequalities for convex bodies. Equality characterizations in these inequalities are related to a Monge Amp`ere differential ...

Added: December 4, 2021

Geometric Aspects of Functional Analysis, Israel Seminar 2014 Vol. 2116 P. 273-293

We generalize the classical Hardy and Faber-Krahn inequalities to arbitrary functions on a convex body Ω ⊂ R n , not necessarily vanishing on the boundary ∂Ω. This reduces the study of the Neumann Poincar´e constant on Ω to that of the cone and Lebesgue measures on ∂Ω; these may be bounded via the curvature ...

Added: April 13, 2015

Mass transportation functionals on the sphere with applications to the logarithmic Minkowski problem

Moscow Mathematical Journal 2020 Vol. 20 No. 1 P. 67-91

We study the transportation problem on the unit sphere Sn−1 for symmetric probability measures and the cost function c(x,y)=log1⟨x,y⟩.
We calculate the variation of the corresponding Kantorovich functional K and study a naturally associated metric-measure space on Sn−1 endowed with a Riemannian
metric generated by the corresponding transportational potential. We introduce a new transportational functional which minimizers are
solutions to the symmetric log-Minkowski problem and ...

Added: October 9, 2019

Calculus of Variations and Partial Differential Equations 2019 Vol. 58 No. 173 P. 1-33

The multistochastic (n, k)-Monge–Kantorovich problem on a product space ∏ni=1Xi∏i=1nXi is an extension of the classical Monge–Kantorovich problem. This problem is considered on the space of measures with fixed projections onto Xi1×⋯×XikXi1×⋯×Xik for all k-tuples {i1,…,ik}⊂{1,…,n}{i1,…,ik}⊂{1,…,n} for a given 1≤k<n1≤k<n. In our paper we study well-posedness of the primal and the corresponding dual problem. Our central result describes a solution ππ to the following important model ...

Added: October 9, 2019

Theory of Stochastic Processes 2016 Vol. 21(37) No. 2 P. 22-28

We study the Monge--Kantorovich problem with one-dimensional marginals $\mu$ and $\nu$ and
the cost function $c = \min\{l_1, \ldots, l_n\}$
that equals the minimum of a finite number $n$ of affine functions $l_i$
satisfying certain non-degeneracy assumptions. We prove that the problem
is equivalent to a finite-dimensional extremal problem. More precisely, it is shown that the solution is concentrated
on ...

Added: December 30, 2017

Remarks on curvature in the transportation metric / Cornell University. Series math "arxiv.org". 2016.

According to a classical result of E.~Calabi any hyperbolic affine hypersphere endowed with its natural Hessian metric has a non-positive Ricci tensor. The affine hyperspheres can be described as the level sets of solutions to the ``hyperbolic" toric K\"ahler-Einstein equation $e^{\Phi} = \det D^2 \Phi$ on proper convex cones. We prove a generalization of this ...

Added: April 14, 2016

Journal Mathematical Economics, Netherlands 2013 Vol. 49 P. 501-505

The famous Afriat’s theorem from the theory of revealed preferences establishes necessary and sufficient conditions for the existence of utility function for a given set of choices and prices. The result on the existence of a homogeneous utility function can be considered as a particular fact of the Monge–Kantorovich mass transportation theory. In this paper ...

Added: September 27, 2013

Mass transportation functionals on the sphere with applications to the logarithmic Minkowski problem

Mass transportation functionals on the sphere with applications to the logarithmic Minkowski problem / Cornell University. Series arXiv "math". 2018.

We study the transportation problem on the unit sphere Sn−1 for symmetric probability measures and the cost function c(x,y)=log1⟨x,y⟩. We calculate the variation of the corresponding Kantorovich functional K and study a naturally associated metric-measure space on Sn−1 endowed with a Riemannian metric generated by the corresponding transportational potential. We introduce a new transportational functional which minimizers are solutions to the symmetric ...

Added: July 31, 2018

Kyoto Journal of Mathematics 2017 Vol. 57 No. 2 P. 293-324

We consider probability measures on R∞ and study optimal transportation mappings for the case of infinite Kantorovich distance. Our examples include 1) quasi-product measures, 2) measures with certain symmetric properties, in particular, exchangeable and stationary measures. We show in the latter case that existence problem for optimal transportation is closely related to ergodicity of the target measure. ...

Added: December 30, 2017

М.: ВИНИТИ РАН, 2018

Сборник трудов семинара по алгебре и геометрии Самарского государственного университета ...

Added: August 19, 2018

Weak regularity of Gauss mass transport / Cornell University. Series math "arxiv.org". 2009. No. 0904.1852.

Given two probability measures $\mu$ and $\nu$ we consider a mass transportation mapping $T$ satisfying 1) $T$ sends $\mu$ to $\nu$, 2) $T$ has the form $T = \phi \frac{\nabla \phi}{|\nabla \phi|}$, where $\phi$ is a function with convex sublevel sets.
We prove a change of variables formula for $T$. We also establish Sobolev estimates for ...

Added: March 27, 2013

Доклады Академии Наук. Математика 2016 Т. 470 № 2 С. 137-140

В работе получены оценки типа Пуанкаре для логарифмически вогнутой меры $\mu$ на
выпуклом множестве $\Omega$. Для этой цели $\Omega$ наделяется римановой метрикой $g$, в которой
риманово многообразие с мерой $(\Omega, g, \mu)$ имеет неотрицательный тензор Бакри-Эмери и,
как следствие, удовлетворяет неравенству Браскампа-Либа.
Рассмотрены несколько естественных классов метрик (гессиановы, конформные),
каждая из которых дает новые весовые неравенства типа Пуанкаре, Харди, логарифмического ...

Added: December 27, 2016

Bulletin des Sciences Mathematiques 2014 Vol. 138 No. 2 P. 165-198

Given two probability measures μ and ν we consider a mass transportation mapping T satisfying 1) T sends μ to ν, 2) T has the form T=φ∇φ|∇φ|, where φ is a function with convex sublevel sets. We prove a change of variables formula for T. We also establish Sobolev estimates for φ, and a ...

Added: December 23, 2015

Atti della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali, Rendiconti Lincei Matematica E Applicazioni 2007 Vol. 18 P. 179-208

We find sufficient conditions for a probability measure $\mu$ to satisfy an inequality of the type $$ \int_{\R^d} f^2 F\Bigl(\frac{f^2}{\int_{\R^d} f^2 d \mu} \Bigr) d \mu \le C \int_{\R^d} f^2 c^{*}\Bigl(\frac{|\nabla f|}{|f|} \Bigr) d \mu + B \int_{\R^d} f^2 d \mu, $$ where $F$ is concave and $c$ (a cost function) is convex. We show ...

Added: March 27, 2013

Theory of Stochastic Processes 2015 Vol. 20(36) No. 2 P. 54-62

We study the Monge and Kantorovich transportation problems on R∞R∞ within the class of exchangeable measures. With the help of the de Finetti decomposition theorem the problem is reduced to an unconstrained optimal transportation problem on a Hilbert space. We find sufficient conditions for convergence of finite-dimensional approximations to the Monge solution. The result holds, in particular, ...

Added: July 8, 2016

Analysis Mathematica 2017 Vol. 43 No. 1 P. 67-88

According to a classical result of E. Calabi any hyperbolic affine hypersphere endowed with its natural Hessian metric has a non-positive Ricci tensor. The affine hyperspheres can be described as the level sets of solutions to the “hyperbolic” toric Kähler–Einstein equation eΦ = detD2Φ on proper convex cones. We prove a generalization of this theorem by showing ...

Added: April 17, 2017

SIAM Journal on Mathematical Analysis 2020 Vol. 52 No. 4 P. 3666-3696

We construct an explicit solution for the multimarginal transportation problem on the unit cube $[0, 1]^3$ with the cost function $xyz$ and one-dimensional uniform projections. We show that the primal problem is concentrated on a set with a nonconstant local dimension and admits many solutions, whereas the solution to the corresponding dual problem is unique ...

Added: August 21, 2020

Доклады Академии наук 2012 Т. 44 № 2 С. 131-136

Работа связана с изучением соболевской регулярности отображений
оптимальной транспортировки в бесконечномерных пространствах, наделенных гауссовской мерой. Найдены условия принадлежности соболевскому классу для таких отображений. Доказана формула замены переменных. ...

Added: February 19, 2013

Успехи математических наук 2012 Т. 67 № 5 С. 3-110

Дан обзор совеременного состояния исследований, связанных с задачами Монжа и Канторовича оптимальной транспортировки мер. ...

Added: February 26, 2014

The KLS isoperimetric conjecture for generalized Orlicz balls / Cornell University. Series arXiv "math". 2016.

What is the optimal way to cut a convex bounded domain $K$ in Euclidean space $(\Real^n,\abs{\cdot})$ into two halves of equal volume, so that the interface between the two halves has least surface area? A conjecture of Kannan, Lov\'asz and Simonovits asserts that, if one does not mind gaining a universal numerical factor (independent of ...

Added: December 27, 2016