Low dimensional sections versus projections of convex bodies

The structure of low dimensional sections and projections of symmetric convex bodies is studied. For a symmetric convex bodyB ⊂ ℝ ⁿ , inequalities between the smallest diameter of rank ℓ projections ofB and the largest in-radius ofm-dimensional sections ofB are established, for a wide range of sub-proportional dimensions. As an application it is shown that every bodyB in (isomorphic) ℓ-position admits a well-bounded (√n, 1)-mixing operator. Research of this author was partially supported by KBN Grant no. 1 P03A 015 27. This author holds the Canada Research Chair in Geometric Analysis.