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On multistochastic Monge–Kantorovich problem, bitwise operations, and fractals

  • Nikita A. Gladkov
  • Alexander V. KolesnikovEmail author
  • Alexander P. Zimin
Article
  • 67 Downloads

Abstract

The multistochastic (nk)-Monge–Kantorovich problem on a product space \(\prod _{i=1}^n X_i\) is an extension of the classical Monge–Kantorovich problem. This problem is considered on the space of measures with fixed projections onto \(X_{i_1} \times \cdots \times X_{i_k}\) for all k-tuples \(\{i_1, \ldots , i_k\} \subset \{1, \ldots , n\}\) for a given \(1 \le k < n\). In our paper we study well-posedness of the primal and the corresponding dual problem. Our central result describes a solution \(\pi \) to the following important model case: \(n=3, k=2, X_i = [0,1]\), the cost function \(c(x,y,z) = xyz\), and the corresponding two-dimensional projections are Lebesgue measures on \([0,1]^2\). We prove, in particular, that the mapping \((x,y) \rightarrow x \oplus y\), where \(\oplus \) is the bitwise addition (xor- or Nim-addition) on \([0,1] \cong {\mathbb {Z}}_2^{\infty }\), is the corresponding optimal transportation. In particular, the support of \(\pi \) is the Sierpiński tetrahedron. In addition, we describe a solution to the corresponding dual problem.

Notes

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Nikita A. Gladkov
    • 1
  • Alexander V. Kolesnikov
    • 1
    Email author
  • Alexander P. Zimin
    • 1
  1. 1.National Research University Higher School of EconomicsMoscowRussian Federation

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