Functionals on the Spaces of n-Dimensional Convex Bodies

Abstract

In geometry, there are several challenging problems studying numbers associated to convex bodies. For example, the packing density problem, the kissing number problem, the covering density problem, the packing-covering constant problem, Hadwiger's covering conjecture and Borsuk's partition conjecture. They are fundamental and fascinating problems about the same objects. However, up to now, both the methodology and the technique applied to them are essentially different. Therefore, a common foundation for them has been much expected. By treating problems of these types as functionals defined on the spaces of n-dimensional convex bodies, this paper tries to create such a foundation. This article suggests an ideal theoretic structure and a couple of research topics such as supderivatives and integral sums of these functionals which seem to be important. In addition, it proves an inequality between the Hausdorff metric and the Banach-Mazur metric and obtains some estimations on the supderivatives.