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arXiv · 2410.01122

Improved stability versions of the Pr\'ekopa-Leindler inequality

Alessio Figalli·João P. G. Ramos
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Abstract

We consider the problem of stability for the Pr\'ekopa-Leindler inequality. Exploiting properties of the transport map between radially decreasing functions and a suitable functional version of the trace inequality, we obtain a uniform stability exponent for the Pr\'ekopa-Leindler inequality. Our result yields an exponent not only uniform in the dimension but also in the log-concavity parameter $\tau = \min(\lambda,1-\lambda)$ associated with its respective version of the Pr\'ekopa-Leindler inequality. As a further application of our methods, we prove a sharp stability result for log-concave functions in dimension 1, which also extends to a sharp stability result for log-concave radial functions in higher dimensions.

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Alessio Figalli, João P. G. Ramos. 2024-10-01. Improved stability versions of the Pr\'ekopa-Leindler inequality. https://arxiv.org/abs/2410.01122

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