Structural results on convexity relative to cost functions
Mass transportation problems appear in various areas of mathematics, their solutions involving cost convex potentials. Fenchel duality also represents an important concept for a wide variety of optimization problems, both from the theoretical and the computational viewpoints. We drew a parallel to the classical theory of convex functions by investigating the cost convexity and its connections with the usual convexity. We give a generalization of Jensen’s inequality for c-convex functions.
Mathematics Subject Classification26A51
KeywordsCost function cost subdifferential cost convex function Jensen inequality Fenchel transform
Unable to display preview. Download preview PDF.
- 1.Caffarelli L.: Allocation maps with general cost functions. In: Partial Differential Equations and Applications. Lecture Notes in Pure and Appl. Math., vol. 177, pp. 29–35. Dekker, New York (1996)Google Scholar
- 4.Karakhanyan A., Wang X.-J.: The reflector design problem. Int. Congress Chin. Math. II, 1–4 (2007)Google Scholar
- 6.Mitroi, F.-C., Niculescu, C.P.: An extension of Young’s inequality. Abstract and Applied Analysis. Article ID 162049. doi:10.1155/2011/162049
- 8.Niculescu C.P., Persson L.-E.: Convex Functions and Their Applications. A Contemporary Approach. CMS Books in Mathematics, vol. 23. Springer, New York (2006)Google Scholar
- 9.Rachev S.T., Rüschendorf L.: Mass Transportation Problems. Probab. Appl. Springer, New York (1998)Google Scholar
- 14.Villani, C.: Optimal Transport. Old and New. Series: Grundlehren der mathematischen Wissenschaften, vol. 338 (2009)Google Scholar