Let $G_{k,n}$ be the grassmannian of $k$-dimensional vector spaces of $\mathbb R^n$. By the Courant-Fisher characterization, the $k$ eigenvalue of an $n \times n$ psd matrix $A$ is given by

$$ \tag{1} \lambda_k = \min_{V \in G_{n-k+1,n}} R(A,V), $$ where $R(A,V):= \max_{x \in V,\,\|x\| = 1} x^\top A x$.

Now, let $V$ be drawn according to the Haar distribution on $G_{k,n}$, and replace the min in (1) by expectation over $V$.

Question. In terms of $k$ and $A$, what does $\mathbb E_V [R(A,V)]$ correspond / evaluate to ?