# Gaussian width (or metric entropy) for the intersection of the $\ell_1$ and $\ell_2$ balls

Let $B_p := \{ x \in \mathbb{R}^d:\; \|x\|_p \le 1\}$ where $\|x\|_p := (\sum_{i=1}^d |x_i|^p)^{1/p}$ is the $\ell_p$ norm.

(1) Let $t \in (0,1)$. Can we give an estimate on $$\mathbb{E} \Big[\sup_{\|x\|_1 \le \frac{1}{1-t}, \; \|x\|_2 \le \frac1{t} } \langle x,w\rangle \Big]$$ where $w$ has standard $d$-dimensional Gaussian distribution? More specifically, can we choose $t = t_n$ such that this scales faster, as $n \to \infty$, than either of the cases $t = 0$ or $t=1$?

Original question: What is the metric entropy of $B_1 \cap (t B_2)$ (in $\ell_2$ norm) for values of $t$ for which the intersection is nontrivial? In particular, how does it depend on $t$?

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Your question lacks a lot of details: you should precise which distance is used on $\mathbb{R}^d$, since at least two come to mind here, and you should precise up to what precision you want to determine metric entropy $H(\varepsilon)= log$ of the minimal size of an $\varepsilon$ covering as $\varpesilon\to0$. Anyway, it is classical and quite obvious that whatever $t$ and reasonable distance, $H(\varepsilon)\sim -d\log(\varepsilon)$. The second term in the asymptotic expansion is a constant, but it is unknown at least for $n>3$ when the distance is Euclidean. – Benoît Kloeckner Mar 25 '13 at 9:00
Thanks. You are right. I forgot to mention the distance, and I am interested in the dependence on $t$ too. – passerby51 Mar 25 '13 at 12:04