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 ${\mathbb{R}}^n$ such that the ${\ell }_p$ distance $(1<p<\infty )$ 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 ${\ell }_p$ sums of Euclidean spaces, deriving upper bounds on the size of an equilateral set for when $p=\infty$, $p$ is even, and for any $1\le p<\infty$.

16 pages; fixed minor issue in proof of Theorem 3

References in corpus (4)

• Equilateral sets in finite-dimensional normed spaces
• Combinatorial distance geometry in normed spaces
• A problem of Kusner on equilateral sets
• Spaces of convex n-partitions