Few distance sets in $\ell_p$ spaces and $\ell_p$ product spaces

Abstract

Kusner asked if n + 1 points is the maximum number of points in R n such that the ℓp distance between any two points is 1. We present an improvement to the best known upper bound when p is large in terms of n, as well as a generalization of the bound to s-distance sets. We also study equilateral sets in the ℓp sums of Euclidean spaces, deriving upper bounds on the size of an equilateral set for when p = ∞, p is even, and for any 1 ≤ p < ∞.