The radii of sections of origin-symmetric convex bodies and their applications (original) (raw)

Let V and W be any convex and origin-symmetric bodies in R n. Assume that for some A ∈L (R n → R n), det A ̸ = 0, V is contained in the ellipsoid A −1 B n (2) , where B n (2) is the unit Euclidean ball. We give a lower bound for the W-radius of sections of A −1 V in terms of the spectral radius of A * A and the expectations of ∥ • ∥ V and ∥ • ∥ W o with respect to Haar measure on S n−1 ⊂ R n. It is shown that the respective expectations are bounded as n → ∞ in many important cases. As an application we offer a new method of evaluation of n-widths of multiplier operators. As an example we establish sharp orders of n-widths of multiplier operators Λ : L p (M d) → L q (M d) , 1 < q ≤ 2 ≤ p < ∞ on compact homogeneous Riemannian manifolds M d. Also, we apply these results to prove the existence of flat polynomials on M d .