caa a22 4500 605496730 CHVBK 20210128100537.0 cr unu---uuuuu 210128e20150401xx s 000 0 eng 10.1007/s10231-013-0376-0 doi (NATIONALLICENCE)springer-10.1007/s10231-013-0376-0 Equality between Monge and Kantorovich multimarginal problems with Coulomb cost [Elektronische Daten] [Maria Colombo, Simone Di Marino] A standard question arising in optimal transport theory is whether the Monge problem and the Kantorovich relaxation have the same infimum; the positive answer means that we can pass to the relaxed problem without loss of information. In the classical case with two marginals, this happens when the cost is positive, continuous, and possibly infinite and the first marginal has no atoms. We study a similar multimarginal symmetric problem, arising naturally in density functional theory, motivated by a recent paper by Buttazzo, De Pascale, and Gori Giorgi. The cost is the potential interaction between n charged particles (hence, it is symmetric, positive, continuous, and infinite whenever $$x_i=x_j$$ x i = x j ), and the marginals are all equal with no atoms. We prove that also in this case, there is equality between the infimum in the cyclical Monge problem (the natural Monge problem in this context) and in the classical Kantorovich problem. This result is new even for 2 marginals, because we consider only transport maps which are involutions. The result is generalized to every symmetric continuous cost function on a Polish space. Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag Berlin Heidelberg, 2013 Optimal transport nationallicence Monge-Kantorovich problem nationallicence Multimarginal problem nationallicence Colombo Maria Scuola Normale Superiore, Pisa, Italy aut Di Marino Simone Scuola Normale Superiore, Pisa, Italy aut Annali di Matematica Pura ed Applicata (1923 -) Springer Berlin Heidelberg 194/2(2015-04-01), 307-320 0373-3114 194:2<307 2015 194 10231 https://doi.org/10.1007/s10231-013-0376-0 text/html Onlinezugriff via DOI BK010053 XK010053 XK010000 Metadata rights reserved Springer special CC-BY-NC licence nationallicence 1 research-article jats NATIONALLICENCE NATIONALLICENCE NL-springer NATIONALLICENCE 856 40 https://doi.org/10.1007/s10231-013-0376-0 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Colombo Maria Scuola Normale Superiore, Pisa, Italy aut NATIONALLICENCE 700 1- Di Marino Simone Scuola Normale Superiore, Pisa, Italy aut NATIONALLICENCE 773 0- Annali di Matematica Pura ed Applicata (1923 -) Springer Berlin Heidelberg 194/2(2015-04-01), 307-320 0373-3114 194:2<307 2015 194 10231