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## An Explicit Solution for a Multimarginal Mass Transportation Problem

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

Gladkov N., Zimin A.

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 (up to addition of constants).

Publication based on the results of:

Колесников А., 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 <img />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 ...

Added: December 23, 2015

Kolesnikov A., 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

Gladkov N., Kolesnikov A., Zimin A., The multistochastic Monge-Kantorovich problem / Cornell University. Series arXiv "math". 2020.

The multistsochastic Monge--Kantorovich problem on the product $X = \prod_{i=1}^n X_i$ of $n$ spaces is a generalization of the multimarginal Monge--Kantorovich problem. For a given integer number $1 \le k<n$ we consider the minimization problem $\int c d \pi \to \inf$ of the space of measures with fixed projections onto every $X_{i_1} \times \dots \times ...

Added: August 21, 2020

Kolesnikov A., 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

Kolesnikov A., 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 ...

Added: February 24, 2016

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

Kolesnikov A., 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

Teplitskaya Y., Степанов Е. О., Paolini E., Advances in Calculus of Variations 2015 Vol. 8 No. 3 P. 267-290

We construct an example of a Steiner tree with an infinite number of branching points connecting an uncountable set of points. Such a tree is proven to be the unique solution to a Steiner problem for the given set of points. As a byproduct we get the whole family of explicitly defined finite Steiner trees, ...

Added: January 31, 2018

Kolesnikov A., Zaev D., 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

Kolesnikov A., Kudryavtseva O., Nagapetyan T., Remarks on the Afriat's theorem and the Monge-Kantorovich problem / Cornell University. Series math "arxiv.org". 2013.

The classical concept of the revealed preferences was introduced by P. Samuelson and studied by H.S. Houthakker, M. Richter, S. Afriat, H. Varian and many others. It was shown by Afriat that the so called SARP (or cyclically consistence) axiom is a necessary and sufficient condition for existence of an appropriate concave utility function for ...

Added: February 23, 2013

Kolesnikov A., Milman E., 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

Kolesnikov A., Theory of Probability and Its Applications 2013 Vol. 57 No. 2 P. 243-264

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 ${\bf R}^d$. 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 ...

Added: December 23, 2015

Kolesnikov A., Zaev D., Exchangeable optimal transportation and log-concavity / Cornell University. Series arXiv "math". 2015.

We study the Monge and Kantorovich transportation problems on 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 the Hilbert space. We find sufficient conditions for convergence of finite-dimensional approximations to the Monge solution. The result holds, in particular, ...

Added: February 23, 2016

Kolesnikov A., Lysenko N. Y., Remarks on mass transportation minimizing expectation of a minimum of affine functions / Cornell University. Series arXiv "math". 2015.

We study Monge-Kantorovich problem with one-dimensional marginals μ,ν and the cost function c=min{l1,…,ln} which equals to minimum of a finite number n of affine functions li 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 the union of n products Ii×Ji, where {Ii}, {Ji} are partitions of the line into unions ...

Added: February 23, 2016

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

Kolesnikov A., 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

Kolesnikov A., Zaev D., Optimal transportation of processes with infinite Kantorovich distance. Independence and symmetry. / Cornell University. Series math "arxiv.org". 2013.

We consider probability measures on $\mathbb{R}^{\infty}$ and study natural analogs of 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. It turns out that the existence problem for optimal transportation is closely related to various ergodic ...

Added: May 13, 2013

Zaev D., Kolesnikov A., 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

Zaev D., On the Monge-Kantorovich problem with additional linear constraints / Cornell University. Series math "arxiv.org". 2014.

We consider the modified Monge-Kantorovich problem with additional restriction: admissible transport plans must vanish on some fixed functional subspace. Different choice of the subspace leads to different additional properties optimal plans need to satisfy. Our main results are quite general and include several important examples. In particular, they include Monge-Kantorovich problems in the classes of ...

Added: May 14, 2014

Klartag B., Kolesnikov A., 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

Zimin A., Gladkov N., An explicit solution for a multimarginal mass transportation problem / Cornell University. Series arXiv "math". 2018.

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 non-constant local dimension and admits many solutions, whereas the solution to the corresponding dual problem is unique (up to ...

Added: October 10, 2018

Kolesnikov A., Lysenko N. Y., 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

Borzykh D., ЛЕНАНД, 2021

Книга представляет собой экспресс-курс по теории вероятностей в контексте начального курса эконометрики. В курсе в максимально доступной форме изложен тот минимум, который необходим для осознанного изучения начального курса эконометрики. Данная книга может не только помочь ликвидировать пробелы в знаниях по теории вероятностей, но и позволить в первом приближении выучить предмет «с нуля». При этом, благодаря доступности изложения и небольшому объему книги, ...

Added: February 20, 2021

Kotelnikova M. V., Aistov A., Вестник Нижегородского университета им. Н.И. Лобачевского. Серия: Социальные науки 2019 Т. 55 № 3 С. 183-189

The article describes a method that allows to improve the content of disciplines of the mathematical cycle by dividing them into invariant (general) and variable parts. The invariants were identified for such disciplines as «Linear algebra», «Mathematical analysis», «Probability theory and mathematical statistics» delivered to Bachelors program students of economics at several universities. Based on ...

Added: January 28, 2020

В. Л. Попов, Математические заметки 2017 Т. 102 № 1 С. 72-80

Мы доказываем, что аффинно-треугольные подгруппы являются борелевскими подгруппами групп Кремоны. ...

Added: May 3, 2017

Красноярск : ИВМ СО РАН, 2013

Труды Пятой Международной конференции «Системный анализ и информационные технологии» САИТ-2013 (19–25 сентября 2013 г., г.Красноярск, Россия): ...

Added: November 18, 2013