MEAN WIDTH AND DIAMETER OF PROPORTIONAL SECTIONS OF A SYMMETRIC CONVEX BODY

Let K be a symmetric convex body in R. The purpose of this paper is to provide upper and lower bounds for the diameter of a random [λn]–dimensional central section of K, where the proportion λ ∈ (0, 1) is arbitrary but fixed. There are several aspects of our approach to this question that should be clarified right away: (1) We are interested in bounds expressed in terms of average parameters of the body K which can be efficiently computed in a simple way, therefore being useful from the computational geometry point of view. (2) The dimension n is understood to tend to infinity. Then, we say that our bounds hold for a random [λn]–dimensional section of K if they are satisfied by all K∩ξ where ξ is in a subset of the appropriate Grassmannian with Haar probability measure greater than 1− h(λ, n), and h(λ, n)→ 0 as n tends to infinity. (3) We say that our estimates are tight for a class of bodies and a fixed proportion λ if the ratio of our upper and lower bounds depends only on λ. It is clear that one cannot obtain tight bounds for the class of all symmetric convex bodies: it is not hard to describe almost degenerated bodies in R (for example, an ellipsoid with highly incomparable semiaxes) for which the diameter of [λn]–sections does not concentrate around some value. So, it is an important question to see under what conditions on K the estimates obtained by a method are tight.