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arXiv · 2206.14872

On Carlier's inequality

Heinz H. Bauschke·Shambhavi Singh·Xianfu Wang
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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.

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Heinz H. Bauschke, Shambhavi Singh, Xianfu Wang. 2022-06-29. On Carlier's inequality. https://arxiv.org/abs/2206.14872

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