Gaussian width after some linear transformation

Question

The Gaussian width $w(T)$ of a set $T\in \mathbb{R}^n$ is defined as follows: $$ w(T) = \mathbb{E}\sup_{x\in T} \langle g,x\rangle $$ where $g$ is a random normal vector in $\mathbb{R}^n$. The Gaussian width provides implicit information of the geometric set. I was curious about the following question: consider a linear map $A: \mathbb{R}^n \rightarrow \mathbb{R}^k$ and define the image $A(T)$ of $T$ $$ A(T) = \{x = Ay \in \mathbb{R}^k | y\in T \}$$ Is there any relation between the Gaussian width $w\left(A(T)\right)$ and $w(T)$? If there is, how would that depend on $A$? I would appreciate for any related reference.

Answer 1

This is actually a HW problem from Vershynin's book and I found it is easy to solve via Sudakov's inequality.