2018/03/28 by Nikita Gladkov, Alexander V. Kolesnikov, Gladkov, Nikita A. +3 · 1 citation
Mathematics · #05Dxx #49J35 #FOS: Mathematics #Functional Analysis (math.FA) #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.1803.10447
openalex publication_date 2018/03/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The multistochastic (n,k)-Monge--Kantorovich problem on a product space\n\∏i=1n Xi is an extension of the classical Monge--Kantorovich\nproblem. This problem is considered on the space of measures with fixed\nprojections onto Xi1 \× \… \× Xik for all k-tuples\n i1, \…, ik \⊂ 1, \…, n for a given 1 \≤ k < n. In\nour paper we study well-posedness of the primal and the corresponding dual\nproblem. Our central result describes a solution \π to the following\nimportant model case: n=3, k=2, Xi = [0,1], the cost function c(x,y,z) =\nxyz, and the corresponding two--dimensional projections are Lebesgue measures\non [0,1]2. We prove, in particular, that the mapping (x,y) \→ x \⊕ y,\nwhere \⊕ is the bitwise addition (xor- or Nim-addition) on [0,1] \≅\n\ℤ2\∞, is the corresponding optimal transportation. In\nparticular, the support of \π is the Sierpi 'nski tetrahedron. In addition,\nwe describe a solution to the corresponding dual problem.\n