A quantitative stability result for the Prékopa–Leindler inequality for arbitrary measurable functions

Böröczky, Károly (Ifj.) and Figalli, Alessio and Ramos, João P. G. (2024) A quantitative stability result for the Prékopa–Leindler inequality for arbitrary measurable functions. ANNALES DE L INSTITUT HENRI POINCARE-ANALYSE NON LINEAIRE, Online. ISSN 0294-1449

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Abstract

We prove that if a triplet of functions satisfies almost equality in the Pr´ekopa–Leindler inequality, then these functions are close to a common log-concave function, up to multiplication and rescaling. Our result holds for general measurable functions in all dimensions, and provides a quantitative stability estimate with computable constants.

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