On the Brunn-Minkowski inequality for general measures with applications to new isoperimetric-type inequalities

By Galyna V. Livshyts, Arnaud Marsiglietti, Piotr Nayar, Artem Zvavitch. 2015

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In this paper we present new versions of the classical Brunn-Minkowski inequality for different classes of measures and sets. We show that the inequality \[ \mu(\lambda A + (1-\lambda)B)^{1/n} \geq \lambda \mu(A)^{1/n} + (1-\lambda)\mu(B)^{1/n} \] holds true for an unconditional product measure $\mu$ with decreasing density and a pair of unconditional convex bodies $A,B \subset \mathbb{R}^n$. We also show that the above inequality is true for any unconditional $\log$-concave measure $\mu$ and unconditional convex bodies $A,B \subset \mathbb{R}^n$. Finally, we prove that the inequality is true for a symmetric $\log$-concave measure $\mu$ and a pair of symmetric convex sets $A,B \subset \mathbb{R}^2$, which, in particular, settles two-dimensional case of the conjecture for Gaussian measure proposed by R. Gardner and the fourth named author. In addition, we deduce the $1/n$-concavity of the parallel volume $t \mapsto \mu(A+tB)$, Brunn's type theorem and certain analogues of Minkowski first inequality.

Subfields: Applied Mathematics, Pharmacology

View Paper: https://www.semanticscholar.org/paper/0f7728ae06b09b951cdc32360c9a93b8ff84c367

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In Applied Mathematics

  1. Yiming Zhao
  2. Yong Huang
  3. Dongmeng Xi
  4. Georgios Moschidis
  5. Alexandros Eskenazis
  6. Qi-Rui Li
  7. Haodi Chen
  8. Shibing Chen

In Pharmacology

  1. Georgios Moschidis
  2. Dongmeng Xi
  3. Yong Huang
  4. Yiming Zhao
  5. Alexandros Eskenazis

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