Functionals on the spaces of 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. In particular, supderivatives for these functionals will be studied.

Covering functionals of cones and double cones

“…For each pair , of convex bodies in , the Banach–Mazur distance (also called the Asplund metric , cf. [ 7 ]) between them is defined by Then is a compact metric space (cf. [ 8 ] and [ 7 ]).…”

Statistical mechanics guides the motions of cm scale objects

“…Iz(γ 0 , γ 1 , γ 2 , κτ ) = e −ε/κτ dΘ = e −mgh/κτ sin( θ)d θd φ (9) where φ ∈ {0, tan −1 (γ 1 /γ 0 )}, (10) θ ∈ {0, tan −1 (γ 0 /γ 2 cos φ)}, (11) h = l 2 (γ 0 sin θ cos φ + γ 1 sin θ sin φ + γ 2 cos θ) (12) whose detailed derivations can be read from Ref. [8].…”

“…saving the cost of computation and experimental testing. This method may also find application to the packing problems [9][10][11][12]. By pouring objects into a container (without shaking), the packing factor may be difficult to calculate without knowing the probability of each object landing in a particular configuration.…”