The Width Directions Of A Reduced Body

A convex body R in Euclidean spacenE is called reduced if the minimal width(35)(K)(27)(35)(R)for every convex body K ?R different from R.A width direction of a convex body is a direction along which its minimal width is attained.A convex body is called strictly convex if its boundary contains no segments.B.V.Dekster proved that a strictly convex reduced domain is of constant width and that the outer normal direction at a regular point of a reduced convex body innE is a width direction.While M.Lassak proved that the normal direction of an extreme support line(the direction of an extreme ray named in this thesis)of a convex reduced domain at its singular point is a width direction.This thesis focuses on the width directions of a strictly convex reduced body at its singular point.The results obtained here extend M.Lassak’s result to reduced bodies in the n-dimensional spaces.Concretely,we discuss first the properties of normal cones and tangent cones,then we prove that the direction of each extreme ray of the normal cone to a strictly convex reduced body in nE at a boundary point stands for a width direction.