  A finite family $\mathcal{F}$ of $d$-dimensional convex polytopes is called
$k$-neighborly if $d-k\le\textup{dim}(C\cap C')\le d-1$ for any two distinct
members $C,C'\in\mathcal{F}$. In 1997, Alon initiated the study of the general
function $n(k,d)$, which is defined to be the maximum size of $k$-neighborly
families of standard boxes in $\mathbb{R}^{d}$. Based on a weighted count of
vectors in $\{0,1\}^{d}$, we improve a recent upper bound on $n(k,d)$ by Alon,
Grytczuk, Kisielewicz, and Przes\l awski for any positive integers $d$ and $k$
with $d\ge k+2$. In particular, when $d$ is sufficiently large and $k\ge
0.123d$, our upper bound on $n(k,d)$ improves the bound
$\sum_{i=1}^{k}2^{i-1}\binom{d}{i}+1$ shown by Huang and Sudakov exponentially.
  Furthermore, we determine that $n(2,4)=9$, $n(3,5)=18$, $n(3,6)=27$,
$n(4,6)=37$, $n(5,7)=74$, and $n(6,8)=150$. The stability result of Kleitman's
isodiametric inequality plays an important role in the proofs.
