The isoperimetric inequality for the Ky Fan norm

2026/07/30 by Lu\'ıs Daniel Abreu
Mathematics · Physics and Astronomy · #math.FA #math-ph #math.CV #math.MP

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arxiv created 2026/07/30 · arxiv updated 2026/07/31

Abstract

We show that, among all measurable sets Ω⊂ 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 1_Ω 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≤ ∞ .

Citations

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• Norm estimates for selfadjoint Toeplitz operators on the Fock space
• Donoho-Logan Large Sieve Principles for Modulation and Polyanalytic Fock Spaces
• Donoho-Logan large sieve principles for modulation and polyanalytic Fock spaces

Related

• The isoperimetric inequality for partial sums of Toeplitz eigenvalues in the Fock space
• An isoperimetric inequality for Gauss--like product measures
• Isoperimetric inequalities for Schatten norms of Riesz potentials
• On the existence of extremizers for the sum of eigenvalues of Toeplitz operators
• Weinstock inequality in higher dimensions