From Brunn-Minkowski to Pr\'ekopa-Leindler and Borell-Brascamp-Lieb: discrete inequalities
Abstract
We consider a general way to obtain Pr\'ekopa-Leindler and Borell-Brascamp-Lieb type inequalities from Brunn-Minkowski type inequalities and provide numerous examples. We use the same heuristic to prove a discrete version of the Pr\'ekopa-Leindler and Borell-Brascamp-Lieb inequalities for functions over Zd. These are the functional extensions of the discrete Brunn-Minkowski inequality conjectured by Ruzsa and recently established by Keevash, Tiba, and the author.
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