My question is regarding mean widths. For a set $\mathcal{T}$ define the mean width
\begin{align*} \omega(T)=\mathbb{E}_{\mathbf{g}\sim\mathcal{N}(0,\mathbf{I})}\bigg[\underset{\mathbf{u}\in\mathcal{T}}{\sup}\text{ } \mathbf{u}^T\mathbf{g}\bigg]. \end{align*}
My question is what is the mean width of two ellipsoid i.e. mean width of the following set \begin{align*} \mathcal{T}=\mathcal{T}_1\cap\mathcal{T}_2 \end{align*} where \begin{align*} \mathcal{T}_1=\{\mathbf{A}_1\mathbf{u}| \|\mathbf{u}\|_{\ell_2}\le 1\}\quad\text{and}\quad \mathcal{T}_2=\{\mathbf{A}_2\mathbf{u}+\mathbf{c}|\|\mathbf{u}\|_{\ell_2}\le 1\} \end{align*} I'm interested in a good upperbound on $\omega(\mathcal{T})$.
