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

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.

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Correspondence to Alexander V. Kolesnikov.

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The second named author was supported by RFBR Project 17-01-00662 and DFG Project RO 1195/12-1. The article was prepared within the framework of the HSE University Basic Research Program and funded by the Russian Academic Excellence Project ‘5-100’.

Communicated by L. Ambrosio.

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Gladkov, N.A., Kolesnikov, A.V. & Zimin, A.P. On multistochastic Monge–Kantorovich problem, bitwise operations, and fractals. Calc. Var. 58, 173 (2019). https://doi.org/10.1007/s00526-019-1610-4

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