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by A V Kolesnikov

Venue: | MIPT Proc. (2010 |

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Alexander V. Kolesnikov

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...ying (31), as explained above. We will also need the following 1-dimensional version of the Caffarelli contraction theorem (which holds true for optimal transport mappings in any dimension, see [18], =-=[25]-=-). Note, however, that in the one-dimensional case the proof is elementary and relies on explicit formulas. Theorem 7.5. Let T : R→ R be the canonical increasing mapping pushing forward a probability ...

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...y-Lipschitz function F is compactly supported. We observe that for any fixed N , assumption (34) holds true. Indeed, we may apply a refinement of Caffarelli’s contraction theorem [7] which appears in =-=[18]-=-, and obtain from (ii) that for any x ∈ Rn, D2ΦN (x) ≥ 1 N2 · Id. We may therefore apply Proposition 4.2, and conclude that for any N ≥ 1, ∫ M+n (R) F 2dθN − (∫ M+n (R) FdθN )2 ≤ 4 ∫ M+n (R) |∇F |2dθN...

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Alexander V. Kolesnikov, Danila, A. Zaev

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...exist c, C such that 0 < c ≤ g ≤ C; 4) f log f ∈ L1(P ). Then there exists an optimal transportation mapping T transforming µ onto ν. Remark 4.2. It follows from Caffarelli’s contraction theorem (see =-=[16]-=-) that assumption 2) is satisfied if (− log pi(xi)) ′′ ≥ C0, (− log qi(xi))′′ ≤ C1 for some C0, C1 > 0 and every i. Of course, there exist many other examples when this assumption is satisfied. Proof....

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