Let $d$ be a large positive integer and fix $r \ge 0$. Set $S := B_2^n \cap [-r,r]^d$, where $B_2^d$ is the euclidean unit-ball in $\mathbb R^d$. Finally, let $\omega(S)$ be the *Gaussian width* of $S$, defined by

$$
\omega(S) := \mathbb E \sup_{x \in S} x^\top z,
$$

where the expectation is over $z \sim N(0,I_d)$.

>**Question.** What is a good upper-bound for $\omega(T)$, valid for large $d$ ?

**Note.** Using [Proposition 1 of this manuscript][1] with $T = [-1/\sqrt{d},1/\sqrt{d}]^d$ (the convex hull of $m=2^d$ points in $B_2^d$), and $s=1/(r\sqrt{d})$, I'm able to obtain the following upper-bound
$$
\omega(S)  = s\cdot\omega(s B_2^d \cap T) \lesssim r\sqrt{d\log(em)} \land \sqrt{d} = rd \land \sqrt{d}.
$$

Unfortunately, the above bound is not very good for my purposes.


  [1]: https://arxiv.org/pdf/1705.10696.pdf