
Journal of Convex Analysis 30 (2023), No. 2, 499514 Copyright Heldermann Verlag 2023 On Carlier's Inequality Heinz H. Bauschke Department of Mathematics, University of British Columbia, Kelowna, Canada heinz.bauschke@ubc.ca Shambhavi Singh Department of Mathematics, University of British Columbia, Kelowna, Canada sambha@student.ubc.ca Xianfu Wang Department of Mathematics, University of British Columbia, Kelowna, Canada shawn.wang@ubc.ca The FenchelYoung inequality is fundamental in Convex Analysis and Optimization. It states that the difference between certain function values of two vectors and their inner product is nonnegative. Recently, Carlier introduced a very nice sharpening of this inequality, providing a lower bound that depends on a positive parameter. In this note, we expand on Carlier's inequality in three ways. First, a duality statement is provided. Secondly, we discuss asymptotic behaviour as the underlying parameter approaches zero or infinity. Thirdly, relying on cyclic monotonicity and associated Fitzpatrick functions, we present a lower bound that features an infinite series of squares of norms. Several examples illustrate our results. Keywords: Carlier's inequality, cyclic monotonicity, Fenchel conjugate, FenchelYoung inequality, Fitzpatrick function, maximally monotone operator, proximal mapping, resolvent. MSC: 26B25, 47H05; 26D07, 90C25. [ Fulltextpdf (141 KB)] for subscribers only. 