Abstract We establish the existence of a regular functional M-position, in the sense of Pisier, for geometric log-concave functions. This provides a functional analogue of Pisier’s regular M-positions for convex bodies and yields uniform control of covering numbers at all scales. Specifically, we show that every isotropic geometric log-concave function f: R^n [0, ) satisfies, for all t 1, align* &\N (f, t g), N (g, t f^{) \}\! (₍ n/t) \N (f^{, t g), N (g, t f) \}\! (₍ n/t), align* where f^ denotes the Legendre dual of f, (t f) (x) =f (x/t) is the t-homothety of f, and ₍ c (n) ^2, ₍ c n. Our result shows that the isotropic position of a log-concave function already provides an almost 1-regular functional M-position.
Giannopoulos et al. (Tue,) studied this question.