Partitioning Functional of a Class of Convex Bodies

For each n-dimensional real Banach space X, each positive integer m, and each bounded set A⊆X with diameter greater than 0, let βX(A,m) be the infimum of δ∈(0,1] such that A⊆X can be represented as the union of m subsets of A, whose diameters are not greater than δ times the diameter of A. Estimating βX(A,m) is an important part of Chuanming Zong’s quantitative program for attacking Borsuk’s problem. However, estimating the partitioning functionals of general convex bodies in finite dimensional Banach spaces is challenging, so we will begin with the estimation of partitioning functionals for special convex bodies. In this paper, we prove a series of inequalities about partitioning functionals of convex cones. Several estimations of partitioning functionals of the convex hull of (A+u)∪(A−u) and (A+u)∪(−A−u) are also presented, where A⊆Rn−1×{0} is a convex body with the origin o in its interior, and u∈Rn∖(Rn−1×{0}). These results contribute to the study of Borsuk’s problem through Zong’s program.