The radii of sections of origin-symmetric convex bodies and their applications

Abstract

LetVandWbe any convex and origin-symmetric bodies inRn.AssumethatforsomeA∈L(Rn→Rn),detA̸=0,Viscontainedin the ellipsoidA−1Bn(2), whereBn(2)is the unit Euclidean ball. WegivealowerboundfortheW-radiusofsectionsofA−1Vintermsof the spectral radius ofA∗Aand the expectations of∥·∥Vand∥·∥Wowith respect to Haar measure onSn−1⊂Rn. It is shownthattherespectiveexpectationsareboundedasn→∞inmanyimportant cases. As an application we offer a new method ofevaluation ofn-widths of multiplier operators. As an examplewe establish sharp orders ofn-widths of multiplier operatorsΛ:Lp(Md)→Lq(Md), 1<q≤2≤p<∞on compacthomogeneous Riemannian manifoldsMd. Also, we apply theseresults to prove the existence of flat polynomials onMd.