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Existence of optimal transport maps for crystalline norms. (English) Zbl 1076.49022

The optimal transport problem (posed by G. Monge in 1781) consists in transporting a given distribution of matter (for instance, a pile of sand) from one place to another minimizing the cost of transportation. If \(\mu, \nu\) are probability measures describing the initial and final distributions, the cost is \[ \int_{R^n} c(x, t(x)) \mu(dx) \] where \(c(x, y)\) is the cost of transporting a unit mass from \(x\) to \(y\) and \(t\) is a transport (a Borel map \(t : R^n \to R^n\) such that \(\mu(t^{-1}(e)) = \nu(e)\) for every Borel set \(e).\) The Monge problem was brought back to life by L. V. Kantorovich in 1942, who, among many other things, introduced the notion of weak (measure-valued) solution. Interest on the Monge-Kantorovich problem has sharply increased in the last two decades.
As the authors point out, existence of optimal transports when \(c(x, y) = h(y - x)\) depends on the strict convexity of \(h,\) thus even the function originally used by Monge \(h(x) = | x| \) (Euclidean norm) is left out. This paper presents an existence proof for crystalline norms; these are given by \[ \| u\| = \max\{ | u\cdot v_i| \, ; \;i = 1, \dots, n\} \, , \] for vectors \(v_1, \dots, v_n\) spanning \(R^n.\)

MSC:

49Q20 Variational problems in a geometric measure-theoretic setting
49J45 Methods involving semicontinuity and convergence; relaxation
28A50 Integration and disintegration of measures
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References:

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