A quantitative stability result for the Pr\'ekopa-Leindler inequality for arbitrary measurable functions
K\'aroly J. B\"or\"oczky, Alessio Figalli, Jo\~ao P. G. Ramos
TL;DR
This paper establishes a quantitative stability estimate for the Prékopa-Leindler inequality, showing that functions nearly satisfying the inequality are close to a common log-concave function, applicable in all dimensions.
Contribution
It provides the first general stability result for the Prékopa-Leindler inequality for arbitrary measurable functions with explicit constants.
Findings
Functions nearly satisfying the inequality are close to a log-concave function.
The stability estimate is quantitative with computable constants.
Applicable in all dimensions for general measurable functions.
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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