  A set in $\mathbb R^d$ is called almost-equidistant if for any three distinct
points in the set, some two are at unit distance apart. First, we give a short
proof of the result of Bezdek and L\'angi claiming that an almost-equidistant
set lying on a $(d-1)$-dimensional sphere of radius $r$, where $r<1/\sqrt{2}$,
has at most $2d+2$ points. Second, we prove that an almost-equidistant set $V$
in $\mathbb R^d$ has $O(d)$ points in two cases: if the diameter of $V$ is at
most $1$ or if $V$ is a subset of a $d$-dimensional ball of radius at most
$1/\sqrt{2}+cd^{-2/3}$, where $c<1/2$. Also, we present a new proof of the
result of Kupavskii, Mustafa and Swanepoel arXiv:1708.01590 that an
almost-equidistant set in $\mathbb R^d$ has $O(d^{4/3})$ elements.
