On Carlier's inequality

TL;DR

This paper extends Carlier's sharpening of the Fenchel-Young inequality by providing a duality perspective, analyzing asymptotic behavior, and introducing a new lower bound involving an infinite series, supported by illustrative examples.

Contribution

It introduces a duality formulation, explores asymptotic limits, and proposes a novel lower bound with an infinite series, expanding the understanding of Carlier's inequality.

Findings

Duality statement for Carlier's inequality

Asymptotic analysis as the parameter approaches zero or infinity

New lower bound involving an infinite series of squared norms

Abstract

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.