### Article

## Remarks on Afriat's theorem and the Monge-Kantorovich problem

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 we explain this viewpoint and discuss some related questions.

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 the union of $n$ products $I_i \times J_i$, where $\{I_i\}$ and $\{J_i\}$

are partitions of the real line into unions of disjoint connected sets.

The families of sets $\{I_i\}$ and $\{J_i\}$ have the following properties: 1) $c=l_i$ on $I_i \times J_i$,

2) $\{I_i\}, \{J_i\}$ is a couple of partitions solving an auxiliary $n$-dimensional extremal problem.

The result is partially generalized to the case of more than two marginals.

This article gives a survey of recent research related to the Monge-Kantorovich problem. Principle results are presented on the existence of solutions and their properties both in the Monge optimal transportation problem and the Kantorovich optimal plan problem, along with results on the connections between both problems and the cases when they are equivalent. Diverse applications of these problems in non-linear analysis, probability theory, and differential geometry are discussed.

We provide a review of methods for studying economic data and building economic indices based on the generalized nonparametric method for computing Konus-Divisia indices, with the focus on the case when the observed behavior is not consistent with the classical Pareto's model of a single rational representative consumer, but may be consistent with the generalized model with two or more rational representative consumers. Computing economic indices allows one to aggregate economic prices and consumption data from the level of individual goods to the level of some general categories of goods.

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. In particular, we prove existence of the symmetric optimal transportation for a certain class of stationary Gibbs measures.

We consider certain spaces of functions on the circle, which naturally appear in harmonic analysis, and superposition operators on these spaces. We study the following question: which functions have the property that each their superposition with a homeomorphism of the circle belongs to a given space? We also study the multidimensional case.

We consider the spaces of functions on the m-dimensional torus, whose Fourier transform is p -summable. We obtain estimates for the norms of the exponential functions deformed by a C1 -smooth phase. The results generalize to the multidimensional case the one-dimensional results obtained by the author earlier in “Quantitative estimates in the Beurling—Helson theorem”, Sbornik: Mathematics, 201:12 (2010), 1811 – 1836.

We consider the spaces of function on the circle whose Fourier transform is p-summable. We obtain estimates for the norms of exponential functions deformed by a C1 -smooth phase.