The John inclusion for log-concave functions

John's inclusion states that a convex body in R d can be covered by the d -dilation of its maximal volume ellipsoid. We obtain a certain John-type inclusion for log-concave functions. As a byproduct of our approach, we establish the following asymptotically tight inequality: For any log-concave function f with finite, positive integral, there exist a positive definite matrix A , a point a ∈ R d , and a positive constant α such that χ B d ( x ) ≤ α f ( A ( x − a ) ) ≤ d + 1 ⋅ e − | x | d + 2 + ( d + 1 ) , where χ B d is the indicator function of the unit ball B d .