Sharp quantitative stability for the Borell-Brascamp-Lieb inequality (and hence Prékopa-Leindler) is proven: near-equality of the integral implies an O(√δ) L1-distance to a p-concave function.
Lower bounds for the Pr \'e kopa-Leindler deficit by some distances modulo translations
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Sharp Quantitative Stability for the Pr\'ekopa-Leindler and Borell-Brascamp-Lieb Inequalities
Sharp quantitative stability for the Borell-Brascamp-Lieb inequality (and hence Prékopa-Leindler) is proven: near-equality of the integral implies an O(√δ) L1-distance to a p-concave function.