Geominimal surface areas in the Brunn-Minkowski-Firey theory.

摘要

In the early 60's W. J. Firey introduced L_p combinations of convex bodies. In the early 90's E. Lutwak introduced L_p curvature of convex bodies. As a result it became possible to extend many of the notions of the classical Brunn-Minkowski theory to an L_p version of this theory (which Lutwak has named the Brunn-Minkowski-Firey theory). One of the important functionals of the Brunn-Minkowski theory is geominimal surface area. This functional was introduced by C. M. Petty in the 70's and serves as a bridge connecting Minkowskian, convex, and affine differential geometry. It is also an important functional in the study of affine isoperimetric inequalities. When extending notions from the classical Brunn-Minkowski theory to the L _p version of the theory there are often a number of possible extensions. In the case of geominimal surface area there are at least two choices. One of these is the subject of this thesis.