Journal of Convex Analysis 32 (2025), No. 1, 291--320
Copyright Heldermann Verlag 2025
Improved Stability Versions of the Prékopa-Leindler Inequality
Alessio Figalli
Department of Mathematics, ETH Zurich, Switzerland
alessio.figalli@math.ethz.ch
Joao P. G. Ramos
Department of Mathematics, ETH Zurich, Switzerland
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 τ = min( λ , 1 - λ) 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.
Keywords: Prekopa-Leindler, stability, log-concavity.
MSC: 26D15; 52A40.
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