doi:10.1007/s12220-017-9858-4engAlonso Gutiérrez, DavidGonzález Merino, BernardoJiménez, Carlos HugoVilla RafaelJohn's ellipsoid and the integral ratio of a log-concave functionART-2018-98820We extend the notion of John’s ellipsoid to the setting of integrable logconcave 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.2018http://zaguan.unizar.es/record/7951010.1007/s12220-017-9858-4http://zaguan.unizar.es/record/79510oai:zaguan.unizar.es:79510info:eu-repo/grantAgreement/ES/MICINN/MTM2013-42105-Pinfo:eu-repo/grantAgreement/ES/MICINN/MTM2016-77710-PJOURNAL OF GEOMETRIC ANALYSIS 28, 2 (2018), 1182-1201All rights reservedhttp://www.europeana.eu/rights/rr-f/info:eu-repo/semantics/openAccess