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In geometry, there are several challenging problems studying numbersassociated to convex bodies. For example, the packing density problem, thekissing number problem, the covering density problem, the packing-coveringconstant problem, Hadwiger's covering conjecture and Borsuk's partitionconjecture. They are fundamental and fascinating problems about the sameobjects. However, up to now, both the methodology and the technique applied tothem are essentially different. Therefore, a common foundation for them hasbeen much expected. By treating problems of these types as functionals definedon the spaces of n-dimensional convex bodies, this paper tries to create such afoundation. This article suggests an ideal theoretic structure and a couple ofresearch topics such as supderivatives and integral sums of these functionalswhich seem to be important. In addition, it proves an inequality between theHausdorff metric and the Banach-Mazur metric and obtains some estimations onthe supderivatives.

Functionals on the Spaces of n-Dimensional Convex Bodies