The isoperimetric inequality for the Ky Fan norm

We show that, among all measurable sets $\Omega \subset \mathbb{C}$ with finite area $s$, the disc of area $s$ maximizes the Ky Fan norm, which is defined as the sum of the first $N$ eigenvalues of the Toeplitz operator with symbol $\mathbf{1}_{\Omega }$ on the Fock space. For $N=1$ this reduces to Nicola-Tilli's celebrated Faber--Krahn inequality and for general $N$ the result was conjectured by Nicola, Riccardi and Tilli, who proved it for radial sets. The proof combines Ky Fan's maximum principle with methods from quantum information theory based on Fock rearrangements of density operators. As a by-product, we obtain isoperimetric inequalities for the Schatten sums in the range $0<p\leq \infty $.