Journal of Convex Analysis 30 (2023), No. 2, 499--514
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 Fenchel-Young 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, Fenchel-Young inequality, Fitzpatrick function, maximally monotone operator, proximal mapping, resolvent.
MSC: 26B25, 47H05; 26D07, 90C25.
[ Fulltext-pdf (141 KB)] for subscribers only.