Remarks on the conjectured log-Brunn–Minkowski inequality

By Christos Saroglou. 2013

120 citations recorded by the source corpus.

Böröczky, Lutwak, Yang and Zhang recently conjectured a certain strengthening of the Brunn-Minkowski inequality for symmetric convex bodies, the so-called log-Brunn-Minkowski inequality. We establish this inequality together with its equality cases for pairs of convex bodies that are both unconditional with respect to some orthonormal basis. Applications of this fact are discussed. Moreover, we prove that the log-Brunn-Minkowski inequality is equivalent to the (B)-Theorem for the uniform measure of the cube (this has been proven by Cordero-Erasquin, Fradelizi and Maurey for the gaussian measure instead).

Subfields: Applied Mathematics, Pharmacology

View Paper: https://www.semanticscholar.org/paper/399bca0817354ceb6b805f0d49d42bc106a7d066

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In Applied Mathematics

1. Emanuel Milman
2. A. V. Kolesnikov
3. Kolesnikov Alexander
4. Milman Emanuel
5. Galyna V. Livshyts
6. Andrea Colesanti
7. Arnaud Marsiglietti
8. Artem Zvavitch

In Pharmacology

1. Galyna V. Livshyts
2. Kolesnikov Alexander
3. Milman Emanuel
4. A. V. Kolesnikov
5. Arnaud Marsiglietti
6. Artem Zvavitch
7. Emanuel Milman
8. Piotr Nayar

Citations in the corpus

• Local Lp-Brunn-Minkowski inequalities for p<1 (2017, background): “iglietti [13] verified the conjecture locally for small-enough C2-perturbations of the Euclidean ball Bn 2 (see below for more details). • The log-Brunn–Minkowski conjecture has been shown by Saroglou [58, 59] to be intimately related to the generalized B-conjecture. Further connections to a conjectu”
• On the Brunn-Minkowski inequality for general measures with applications to new isoperimetric-type inequalities (2015, methodology): “In [31] the proof is done by linking the problem to the new logBrunn-Minkowski inequality of Böröczky, Lutwak, Yang and Zhang; see [6], [7], [30] and [31].”
• Local $L^p$-Brunn-Minkowski inequalities for $p<1$ (2017, background): “• The log-Brunn–Minkowski conjecture has been shown by Saroglou [58, 59] to be intimately related to the generalized B-conjecture.”
• On the stability of Brunn–Minkowski type inequalities (2016, background): “The latter is indeed true: it follows from the results of [11] and [23].”