Improved stability versions of the Prékopa-Leindler inequality

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.

28 pages; In honor of R. T. Rockafellar, for his 90th birthday