On the Dual Form of “low M∗-estimate” in the Quasi-convex Case

Let B 2 denote the Euclidean ball in IR , and, given closed star-shaped body K ⊂ IR, MK denote the average of the gauge of K on the Euclidean sphere. Let p ∈ (0, 1) and let K ⊂ IR be a p-convex body. In [17] we proved that for every λ ∈ (0, 1) there exists an orthogonal projection P of rank (1− λ)n such that f(λ) MK PB 2 ⊂ PK, where f(λ) = cpλ for some positive constant cp depending on p only. In this note we prove that f(λ) can be taken equal to Cpλ. In terms of Kolmogorov numbers it means that for every k ≤ n dk(Id : `2 −→ (IR, ‖ · ‖K)) ≤ Cp n1/p−1 k1/p−1/2 `(Id : `2 −→ (IR, ‖ · ‖K)), where `(Id) = E‖ ∑n i=1 giei‖K for the independent standard Gaussian random variables {gi} and the canonical basis {ei} of IR. All results do not require the symmetry of K.