Let $H$ be a separable Hilbert space, $\Pi:H\to L$ the orthogonal projection to a linear subspace of finite dimension $p$, and $U$ the open cone of vectors $u\in H$ such that $\langle u,\Pi u\rangle<\|u\|^2/\sqrt{2}$. Which is the maximum number of orthonormal vectors contained in $U$?