A version of Urysohn's inequality states that for a symmetric convex body $K \subset \mathbb{R}^n$, one has $$ \left(\frac{\text{vol}(K)}{\text{vol}(B_2)} \right)^{1/n} \le \frac{1}{\sqrt{n}} E \; \| g \|_{K^\circ} $$ where $g$ is a canonical (or isotropic) Gaussian vector ($g \sim N(0,I_n$)) and $K^\circ$ is the polar body of $K$, or equivalently, $$\| g\|_{K^\circ} := \sup_{y \in K} {\langle x,y\rangle} $$

$E \|g\|_{K^\circ}$ is usually called the Gaussian width of $K$.

Are there extensions of this result to more general sets $K$? For example, it seems easy to replace $K$ with the intersection of $K$ and a lower dimensional subspace of $\mathbb{R}^n$. How about the case where we replace $K$ with $K \cap (\cup_{i \in I} S_i)$ where each $S_i$ is a subspace (and $I$ is finite)? In general what is the most general form known?

A version of Urysohn's inequality states that for a symmetric convex body $K \subset \mathbb{R}^n$, one has $$ \left(\frac{\text{vol}(K)}{\text{vol}(B_2)} \right)^{1/n} \le \frac{1}{\sqrt{n}} E \; \| g \|_{K^\circ} $$ where $g$ is a canonical (or isotropic) Gaussian vector ($g \sim N(0,I_n$)) and $K^\circ$ is the polar body of $K$, or equivalently, $$\| g\|_{K^\circ} := \sup_{y \in K} {\langle x,y\rangle} $$

$E \|g\|_{K^\circ}$ is usually called the Gaussian width of $K$.

Are there extensions of this result to more general sets $K$? For example, it seems easy to replace $K$ with the intersection of $K$ and a lower dimensional subspace of $\mathbb{R}^n$. How about the case where we replace $K$ with $K \cap (\cup_{i \in I} S_i)$ where each $S_i$ is a subspace (and $I$ is finite)? What is the most general form known?