John's ellipsoid and the integral ratio of a log-concave function

2015/11/04 by Alonso-Gutiérrez, David, Merino, Bernardo González, Jiménez, Carlos Hugo +1 · 2 citations
#FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1511.01266

Abstract

We extend the notion of John's ellipsoid to the setting of integrable log-concave functions. This will allow us to define the integral ratio of a log-concave function, which will extend the notion of volume ratio, and we will find the log-concave function maximizing the integral ratio. A reverse functional affine isoperimetric inequality will be given, written in terms of this integral ratio. This can be viewed as a stability version of the functional affine isoperimetric inequality.

Cited by

• On local Liakopoulos-Meyer type inequalities and their functional counterparts
• On stochastic forms of functional isoperimetric inequalities

Related

• L_p John ellipsoids for log-concave functions
• Functional John Ellipsoids
• Functional Löwner Ellipsoids
• An affine isoperimetric inequality for log-concave functions
• The John inclusion for log-concave functions