Notes On The Widths In Gromov's Sense

In his a classical paper about metric geometry:Width and related invari-ants of Riemannian Mandifolds,Asterisque,1988,163-164:93-109,M.Gromov introduced the notions of the k-width of a subset in Euclidean space,and the k-diameter of a metric space.Gromov pointed out an inequality relation between the k-widths and the volume without the details of proof.In this paper,we fill these details of proof,and improve the corresponding constants.In addition,we prove that the k-diameter of a product space with the natural metric can be dominated by the s-diameters of its factor spaces(s?k)We also study the upper and bounds of k-diameters of a compact convex hypersurface and compact convex set in Eudidean space.This paper consists of six sections.In section 1,we introduce the main results of the paper.In section 2,we define k-width and k-diameter,and give some basic facts on geometry.In section 3,we compute the k-width of an ellipsoid.In section 4,by approximating a convex body by simplex and ellipsoid,we estimate the volume of a convex body.In section 5,we discuss the k-diameter of a product space.In section 6,we prove the two theorems about the widths of compact convex hyper surface and compact convex set.