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 τ=min⁡(λ,1−λ)\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.

Author information

Contact details are reproduced from the original publication and may be historical.

João P. G. Ramos

Department of Mathematics, ETH Zurich, Switzerland

Keywords

  • Prekopa-Leindler
  • stability
  • log-concavity

Mathematics Subject Classification

26D15; 52A40

References

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Suggested citation

A. Figalli, J. P. G. Ramos. “Improved Stability Versions of the Prékopa-Leindler Inequality.” Journal of Convex Analysis 32 (2025), No. 1, 291–320. https://doi.org/10.68381/jca32016

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Published by Heldermann Verlag, 2025. Rights now held by Banach Press LLC.