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

The multistochastic ( n , k )-Monge–Kantorovich problem on a product space ∏ 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 × ⋯ × X i k for all k -tuples { i 1 , … , i k } ⊂ { 1 , … , n } for a given 1 ≤ k < n . In our paper we study well-posedness of the primal and the corresponding dual problem. Our central result describes a solution π to the following important model case: n = 3 , k = 2 , X i = [ 0 , 1 ] , the cost function c ( x , y , z ) = x y z , and the corresponding two-dimensional projections are Lebesgue measures on [ 0 , 1 ] 2 . We prove, in particular, that the mapping ( x , y ) → x ⊕ y , where ⊕ is the bitwise addition (xor- or Nim-addition) on [ 0 , 1 ] ≅ Z 2 ∞ , is the corresponding optimal transportation. In particular, the support of π is the Sierpiński tetrahedron. In addition, we describe a solution to the corresponding dual problem.