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We prove that any Kantorovich potential for the cost function c = d2/2 on a Riemannian manifold (M, g) is locally semiconvex in the “region of interest”, without any compactness assumption on M, nor any assumption on its curvature. Such a region of interest is of full μ-measure as soon as the starting measure μ does not charge n - 1-dimensional rectifiable sets.
@article{COCV_2011__17_3_648_0,
author = {Figalli, Alessio and Gigli, Nicola},
title = {Local semiconvexity of {Kantorovich} potentials on non-compact manifolds},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {648--653},
publisher = {EDP-Sciences},
volume = {17},
number = {3},
year = {2011},
doi = {10.1051/cocv/2010011},
mrnumber = {2826973},
zbl = {1228.49047},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/cocv/2010011/}
}
TY - JOUR AU - Figalli, Alessio AU - Gigli, Nicola TI - Local semiconvexity of Kantorovich potentials on non-compact manifolds JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2011 SP - 648 EP - 653 VL - 17 IS - 3 PB - EDP-Sciences UR - http://geodesic.mathdoc.fr/articles/10.1051/cocv/2010011/ DO - 10.1051/cocv/2010011 LA - en ID - COCV_2011__17_3_648_0 ER -
%0 Journal Article %A Figalli, Alessio %A Gigli, Nicola %T Local semiconvexity of Kantorovich potentials on non-compact manifolds %J ESAIM: Control, Optimisation and Calculus of Variations %D 2011 %P 648-653 %V 17 %N 3 %I EDP-Sciences %U http://geodesic.mathdoc.fr/articles/10.1051/cocv/2010011/ %R 10.1051/cocv/2010011 %G en %F COCV_2011__17_3_648_0
Figalli, Alessio; Gigli, Nicola. Local semiconvexity of Kantorovich potentials on non-compact manifolds. ESAIM: Control, Optimisation and Calculus of Variations, Tome 17 (2011) no. 3, pp. 648-653. doi: 10.1051/cocv/2010011
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