qThe majority of this thesis develops the Minkowski inequalities and considers their interpretation in three-dimensional Euclidean space. In particular, for a given convex body, two cubic inequalities involving the volume, surface area and the integral of mean curvature are formulated. By defining an x-coordinate based upon the surface area and the integral of mean curvature squared and a y-coordinate dependent upon the volume and the cube of the integral of mean curvature, a map from the set of all convex bodies in R3 to the unit square is generated. The image of this map is called the Blaschke diagram. The equality cases from the Minkowski inequalities form one boundary for the diagram. Then appears, however, to be a missing boundary connecting the point (8/[mathematical symbol Pi]2, 0), the image of disks, to the point (1, 1), the image of spheres. The goal of this thesis is to develop the mathematics necessary to discover the convex bodies that form the missing boundary. Ideally, this boundary would correspond to a third inequality between the volume, surface area, and the integral of mean curvature for convex bodiesq--Document.Ideally, this boundary would correspond to a third inequality between the volume, surface area, and the integral of mean curvature for convex bodiesaquot;--Document.
|Title||:||The Blaschke Diagram for Convex Bodies in R3|
|Author||:||Anthony J. Ford|