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

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 $1\le p<\infty$.