On the (fi sub 2-) behaviour of linear functionals on isotropic convex bodies
The slicing problem can be reduced to the study of isotropic convex bodies K with diam(K) ≤ cnLK , where LK is the isotropic constant. We study the ψ2-behavior of linear functionals on this class of bodies. It is proved that ‖〈·, θ〉‖ψ2 ≤ CLK for all θ in a subset U of S with measure σ(U) ≥ 1 − exp(−c√n). However, there exist isotropic convex bodies K with uniformly bounded geometric distance from the Euclidean ball, such that maxθ∈Sn−1 ‖〈·, θ〉‖ψ2 ≥ c 4 √ nLK . In a different direction, we show that good average ψ2-behavior of linear functionals on an isotropic convex body implies very strong dimension-dependent concentration of volume inside a ball of radius r ≃ nLK .