Weighted Brunn-Minkowski Theory II: Inequalities for Mixed Measures and Applications
arxiv(2024)
摘要
In "Weighted Brunn-Minkowski Theory I", the prequel to this work, we
discussed how recent developments on concavity of measures have laid the
foundations of a nascent weighted Brunn-Minkowski theory. In particular, we
defined the mixed measures of three convex bodies and obtained its integral
representation. In this work, we obtain inequalities for mixed measures, such
as a generalization of Fenchel's inequality; this provides a new, simpler proof
of the classical volume case. Moreover, we show that mixed measures are
connected to the study of log-submodularity and supermodularity of the measure
of Minkowski sums of convex bodies. This elaborates on the recent
investigations of these properties for the Lebesgue measure. We conclude by
establishing that the only Radon measures that are supermodular over the class
of compact, convex sets are multiples of the Lebesgue measure. Motivated by
this result, we then discuss weaker forms of supermodularity by restricting the
class of convex sets.
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