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.
Near equality in the two-dimensional Brunn-Minkowski inequality
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abstract
If a pair of subsets of two-dimensional Euclidean space nearly achieves equality in the Brunn-Minkowski inequality, in the sense that the measure of the associated sumset is nearly equal to the lower bound provided by the inequality, then these sets must nearly coincide with a pair of homothetic convex sets. The proof relies on a continuum analogue of a theorem of Freiman which characterizes finite sets of integers whose sumsets are of nearly minimal size. Small corrections and clarifications have been made in this draft.
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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.