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Symmetric Monge–Kantorovich problems and polar decompositions of vector fields
| Content Provider | Semantic Scholar |
|---|---|
| Author | Ghoussoub, Nassif Moameni, Abbas |
| Copyright Year | 2013 |
| Abstract | We address the problem of whether a bounded measurable vector field from a bounded domain Ω into $${\mathbb{R}^d}$$Rd is N-cyclically monotone up to a measure preserving N-involution, where N is any integer larger than 2. Our approach involves the solution of a multidimensional symmetric Monge–Kantorovich problem, which we first study in the case of a general cost function on a product domain ΩN. The polar decomposition described above corresponds to a special cost function derived from the vector field in question (actually N − 1 of them). The problem amounts to showing that the supremum in the corresponding Monge–Kantorovich problem when restricted to those probability measures on ΩN which are invariant under cyclic permutations and with a given first marginal μ, is attained on a probability measure that is supported on a graph of the form x → (x, Sx, S2x,..., SN-1x), where S is a μ-measure preserving transformation on Ω such that SN = I a.e. The proof exploits a remarkable duality between such involutions and those Hamiltonians that are N-cyclically antisymmetric. |
| Starting Page | 1129 |
| Ending Page | 1166 |
| Page Count | 38 |
| File Format | PDF HTM / HTML |
| DOI | 10.1007/s00039-014-0287-2 |
| Alternate Webpage(s) | https://arxiv.org/pdf/1302.2886v2.pdf |
| Alternate Webpage(s) | http://www.birs.ca/~nassif/papers_download/ghoussoub-momeni_10-02-13/ghoussoub-momeni_05-09-13.pdf |
| Alternate Webpage(s) | http://www.birs.ca/~nassif/papers_download/ghoussoub-momeni_10-02-13/ghoussoub-momeni_30-04-14.pdf |
| Alternate Webpage(s) | http://www.birs.ca/~nassif/papers_download/ghoussoub-momeni_10-02-13/ghoussoub-momeni_10-02-13.pdf |
| Alternate Webpage(s) | https://page-one.springer.com/pdf/preview/10.1007/s00039-014-0287-2 |
| Alternate Webpage(s) | https://doi.org/10.1007/s00039-014-0287-2 |
| Volume Number | 24 |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
