IMPROVED LOG-CONCAVITY FOR ROTATIONALLY INVARIANT MEASURES OF SYMMETRIC CONVEX SETS

Dario Cordero-Erausquin; Liran Rotem

doi:10.1214/22-AOP1604

IMPROVED LOG-CONCAVITY FOR ROTATIONALLY INVARIANT MEASURES OF SYMMETRIC CONVEX SETS

Dario Cordero-Erausquin, Liran Rotem

Mathematics

Technion - Israel Institute of Technology

Research output: Contribution to journal › Article › peer-review

Abstract

We prove that the (B) conjecture and the Gardner–Zvavitch conjecture are true for all log-concave measures that are rotationally invariant, extending previous results known for Gaussian measures. Actually, our result apply beyond the case of log-concave measures, for instance, to Cauchy measures as well. For the proof, new sharp weighted Poincaré inequalities are obtained for even probability measures that are log-concave with respect to a rotationally invariant measure.

Original language	English
Pages (from-to)	987-1003
Number of pages	17
Journal	Annals of Probability
Volume	51
Issue number	3
DOIs	https://doi.org/10.1214/22-AOP1604
State	Published - 2023

Keywords

(B) conjecture
Brascamp–Lieb inequality
Brunn–Minkowski
Gardner–Zvavitch conjecture
Poincaré inequality
log-concavity

All Science Journal Classification (ASJC) codes

Statistics and Probability
Statistics, Probability and Uncertainty

Access to Document

10.1214/22-AOP1604

Cite this

@article{9c48306cd2f2445cac4d6d9d64b1b771,

title = "IMPROVED LOG-CONCAVITY FOR ROTATIONALLY INVARIANT MEASURES OF SYMMETRIC CONVEX SETS",

abstract = "We prove that the (B) conjecture and the Gardner–Zvavitch conjecture are true for all log-concave measures that are rotationally invariant, extending previous results known for Gaussian measures. Actually, our result apply beyond the case of log-concave measures, for instance, to Cauchy measures as well. For the proof, new sharp weighted Poincar{\'e} inequalities are obtained for even probability measures that are log-concave with respect to a rotationally invariant measure.",

keywords = "(B) conjecture, Brascamp–Lieb inequality, Brunn–Minkowski, Gardner–Zvavitch conjecture, Poincar{\'e} inequality, log-concavity",

author = "Dario Cordero-Erausquin and Liran Rotem",

note = "Publisher Copyright: {\textcopyright} Institute of Mathematical Statistics, 2023",

year = "2023",

doi = "10.1214/22-AOP1604",

language = "الإنجليزيّة",

volume = "51",

pages = "987--1003",

journal = "Annals of Probability",

issn = "0091-1798",

publisher = "Institute of Mathematical Statistics",

number = "3",

}

TY - JOUR

T1 - IMPROVED LOG-CONCAVITY FOR ROTATIONALLY INVARIANT MEASURES OF SYMMETRIC CONVEX SETS

AU - Cordero-Erausquin, Dario

AU - Rotem, Liran

N1 - Publisher Copyright: © Institute of Mathematical Statistics, 2023

PY - 2023

Y1 - 2023

N2 - We prove that the (B) conjecture and the Gardner–Zvavitch conjecture are true for all log-concave measures that are rotationally invariant, extending previous results known for Gaussian measures. Actually, our result apply beyond the case of log-concave measures, for instance, to Cauchy measures as well. For the proof, new sharp weighted Poincaré inequalities are obtained for even probability measures that are log-concave with respect to a rotationally invariant measure.

AB - We prove that the (B) conjecture and the Gardner–Zvavitch conjecture are true for all log-concave measures that are rotationally invariant, extending previous results known for Gaussian measures. Actually, our result apply beyond the case of log-concave measures, for instance, to Cauchy measures as well. For the proof, new sharp weighted Poincaré inequalities are obtained for even probability measures that are log-concave with respect to a rotationally invariant measure.

KW - (B) conjecture

KW - Brascamp–Lieb inequality

KW - Brunn–Minkowski

KW - Gardner–Zvavitch conjecture

KW - Poincaré inequality

KW - log-concavity

UR - http://www.scopus.com/inward/record.url?scp=85145879392&partnerID=8YFLogxK

U2 - 10.1214/22-AOP1604

DO - 10.1214/22-AOP1604

M3 - مقالة

SN - 0091-1798

VL - 51

SP - 987

EP - 1003

JO - Annals of Probability

JF - Annals of Probability

IS - 3

ER -

IMPROVED LOG-CONCAVITY FOR ROTATIONALLY INVARIANT MEASURES OF SYMMETRIC CONVEX SETS

Abstract

Keywords

All Science Journal Classification (ASJC) codes

Access to Document

Other files and links

Fingerprint

Cite this