We prove the following isoperimetric-type inequality: Given a finite absolutely continuous Borel measure on, half-spaces have maximal measure among all subsets with prescribed barycenter. As a consequence, we make progress towards a solution to a problem of Henk and Pollehn, which is equivalent to a log-Minkowski inequality for a parallelotope and a centered convex body. Our probabilistic approach to the problem also gives rise to several inequalities and conjectures concerning the truncated mean of certain log-concave random variables.
Bibliographical noteFunding Information:
This work was supported by the European Research Council [grant number 637386 to P.H.-K.]; and by the Israel Science Foundation [grant number 784/20 to B.S.]. Acknowledgments
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