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Article

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

Calculus of Variations and Partial Differential Equations. 2019. Vol. 58. No. 173. P. 1-33.
Kolesnikov A., Gladkov N., Zimin A.

The multistochastic (nk)-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 case: n=3,k=2,Xi=[0,1]n=3,k=2,Xi=[0,1], the cost function c(x,y,z)=xyzc(x,y,z)=xyz, and the corresponding two-dimensional projections are Lebesgue measures on [0,1]2[0,1]2. We prove, in particular, that the mapping (x,y)→x⊕y(x,y)→x⊕y, where ⊕⊕ is the bitwise addition (xor- or Nim-addition) on [0,1]≅Z∞2[0,1]≅Z2∞, is the corresponding optimal transportation. In particular, the support of ππ is the Sierpiński tetrahedron. In addition, we describe a solution to the corresponding dual problem.