Erdős distance problems in normed spaces
Computational Geometry: Theory and Applications
European Journal of Combinatorics
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A Minkowski space Md = (Rd, || ||) is just Rd with distances measured using a norm || ||. A norm || || is completely determined by its unit ball {x ∈ Rd | ||x|| ≤ 1} which is a centrally symmetric convex body of the d-dimensional Euclidean space Ed. In this note we give upper bounds for the maximum number of times the minimum distance can occur among n points in Md, d ≥ 3. In fact, we deal with a somewhat more general problem namely, we give upper bounds for the maximum number of touching pairs in a packing of n translates of a given convex body in Ed, d ≥ 3.