Abstract
We prove that under quite general condition on a cost function c in \({\mathbb {R}}^n\) the Hausdorff dimension of the singular set of a c-concave function has dimension at most \(n-1\). Our result applies for non-semiconcave cost functions and has applications in optimal mass transportation.
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References
Alberti, G., Ambrosio, L., Cannarsa, P.: On the singularities of convex functions. Manuscr. Math. 76, 421–436 (1992)
Alberti, G., Ambrosio, L.: A geometrical approach to monotone functions in \(\mathbb{R}^n\). Math. Z 230, 259–316 (1999)
Ambrosio, L., Gigli, N., Savaré, G.: Gradient Flows in Metric Spaces and in the Space of Probability Measures, Lectures in Mathematics. Birkhäuser Verlag, Basel (2005)
Ambrosio, L., Rigot, S.: Optimal mass transportation in Heisenberg group. J. Funct. Anal. 208, 261–301 (2004)
Cannarsa, P., Sinestrari, C.: Semiconcave Functions, Hamilton–Jacobi Equations, and Optimal Control. Birkäuser, Basel (2004)
Figalli, A., Rifford, L.: Mass transportation on sub-Riemannian manifolds. Geom. Funct. Anal. 20, 124–159 (2010)
Fu, J.H.G.: Monge ampere functions. I. Indiana Math. J. 38(3), 745–771 (1989)
Gutierrez, C.E., Huang, Q.: The refractor problem in reshaping light beams. Arch. Ration. Mech. Anal. 193(2), 423–443 (2009)
Gutierrez, C.E., Huang, Q.: The near field refractor. Annales de l’Institut Henri Poincaré 31(4), 655–684 (2014)
Mattila, P.: Distribution of sets and measures along planes. J. Lond. Math. Soc. 38, 125–132 (1988)
Mattila, P.: Geometry of Sets and Measures in Euclidean Spaces. Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (1995)
Reshetnyak, Y.G.: Generalized derivative and differentiability almost everywhere. Math. USSR Sbornik 4, 293–302 (1968)
Rifford, L.: Sub-riemannian Geometry and Optimal Transport. Springer Briefs in Mathematics. Springer, Cham (2014)
Villani, C.: Topics in Optimal Transportation. AMS, Providence (2003)
Villani, C.: Optimal Transport, Old and New, Grundlehren Math. Wiss, vol. 338. Springer, Cham (2008)
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The authors acknowledge the support of the Swiss National Science Foundation Grant No. 200020-146477.
