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](https://arxiv.org/pdf/1705.10696.pdf) 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.



Gaussian width of intersection of cube and ball in high-dimensional euclidean space
The answer is, up to a constant factor $\omega(S) = \Theta(\min(\sqrt{d}, rd))$.


To see the upper bound, we can use the fact that if $S \subset Q$, then $\omega(S) \leq \omega(Q)$, therefore
$$
\omega(B\_2^n \cap r B\_\infty^n) \leq \min(\omega(B\_2^n), r \omega(B\_\infty^n)) = \min( \mathbb{E}\_{z \sim \mathcal{N}(0, I)} \|z\|\_2, r\, \mathbb{E}\_{z \sim \mathcal{N}(0, I)} \|z\|\_1) = \mathcal{O}(\min(\sqrt{d}, rd)).
$$


For the lower bound, note that for any given vector $z$, we can chose $\tilde{z} \in S$ to be $\tilde{z}\_i := \min(1/\sqrt{d}, r) \mathrm{sgn}(z\_i)$. With this choice, we have for any fixed $z$
$$\sup\_{x \in S} x^T z \geq \tilde{z}^T z = \min(1/\sqrt{d}, r) \sum\_i |z\_i| = \min(1/\sqrt{d}, r) \|z\|\_1,$$
and therefore $\omega(S) \geq \min(1/\sqrt{d}, r) \mathbb{E}\_{z\sim\mathcal{N}(0, I)} \|z\|\_1 = \Omega(\min(\sqrt{d}, rd)$.


Note that this is not inconsistent with the upper bound given by Proposition 1 in the cited paper --- the hypercube $[-r, r]^d$ has $2^d$ vertices, not $2d$.


