Let $d < n$, and let $G_n(d)$ denote the space of all $d$-dimensional subspaces of $\mathbb{R}^n$.
Let $a = (a_1,\dots, a_n)$ denote a positive sequence, and define $U(a) = \{u \in \mathbb{R}^n: \sum_{j} u_j^2/a_j \leq 1\}$. Then define $$ \lambda_a(K) = \sup_{u \in K \cap U(a)} \sum_{j=1}^n u_j^2 $$
I am interested in $\mu_a = \mathbb{E}[\lambda_a(K)]$, where the expectation is with respect to $K$ drawn from the uniform (Haar) measure on $G_n(d)$. Is this possible to compute?
Two obvious observations:
- In the special case that $a_1 = a_2 = \cdots= a_d \equiv \alpha$, then evidently $$ \lambda_a(K) = \alpha, $$ with probability $1$. The supremum is attained by taking any unit vector in $K$ and scaling it appropriately.
- In general, $\lambda_a(K) \leq \max_j a_j$, and hence $\mu_a \leq \max_j a_j$.
I have included the tag random-matrices as this problem can equivalently be formulated in terms of the kernel of a random matrix $G$ filled with standard Normal entries.
