Gaussian width of the sphere and of the sparse set

ProofAdvanced⏱ 30 minHigh-Dim Probabilitygaussian widthspheresparsity

Problem

Show that the Gaussian width of the unit sphere $S^{d-1}$ satisfies $w(S^{d-1}) = \E\|g\|_2 \in [\sqrt{d-1},\sqrt d]$ .
Show that for the set of $s$ -sparse unit vectors, $w(\mathcal S_s) \le C\sqrt{s\log(ep/s)}$ .

Write a complete argument. Every inequality you use should be one you could state precisely.

Hints — 0 of 2 revealed

Opening this before you have a real attempt on paper is the fastest way to feel like you understand something you cannot reproduce.

Problem	Field	Type	Difficulty	Time
Simulate the measure of a spherical cap run	High-Dim Probability	Coding	Intermediate	30 min
The geometry of l1 versus l2 constraints	High-Dim Statistics	Proof	Intermediate	30 min
The operator norm on a net	High-Dim Probability	Proof	Intermediate	30 min
The volumetric lower bound on covering numbers	High-Dim Probability	Proof	Intermediate	30 min
Covering numbers of the set of sparse unit vectors	High-Dim Probability	Proof	Advanced	1 hour