Accepted Paper
Inserted: 7 oct 2024
Last Updated: 12 oct 2024
Year: 2024
Abstract:
We consider the problem of stability for the Prékopa–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 Prékopa–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 Prékopa–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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