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

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