Universal Covering System and Borsuk’s Problem in Finite Dimensional Banach Spaces

For each n-dimensional real Banach space X and each positive integer m, let β(X,m) be the infimum of δ∈(0,1] such that each set A⊆X having diameter 1 can be represented as the union of m subsets of A, whose diameters are not greater than δ. Providing accurate estimations of β(X,m) for specific choices of X and m is crucial for addressing the extension of the classical Borsuk’s problem. A general framework for estimating β(X,m) via constructing and refining universal covering systems is presented. As an example, a universal covering system is constructed in ℓ13 and it is shown that β(ℓ13,8)≤11/12 by a feasible partitioning of members in this universal covering system.