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<p>In <i>Weighted Brunn–Minkowski Theory. I</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.</p>

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Weighted Brunn–Minkowski theory. II. Inequalities for mixed measures and applications

  • 期刊论文

  • 发表日期：

    2026年2月24日

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  • Matthieu Fradelizi,
  • Dylan Langharst,
  • Mokshay Madiman,
  • Artem Zvavitch

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.