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The isoperimetric inequality for the Ky Fan norm

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Pith's one-line read Among all finite-area sets in the plane, discs maximize the sum of the first N eigenvalues of the Fock-space Toeplitz operator.

desk verdict Short, clean proof that discs maximize Ky Fan sums of Fock Toeplitz eigenvalues for arbitrary measurable sets, settling the NRT conjecture via De Palma majorization. read the letter →

arxiv 2607.28376 v1 pith:64GYK3QW submitted 2026-07-30 math.FA math-phmath.CVmath.MP

Lu\'{i}s Daniel Abreu This is my paper · ORCID
classification math.FAmath-phmath.CVmath.MP MSC 47B3542C4081S3049Q1046E22
keywords isoperimetricinequalityKyFannormToeplitzoperatorBargmann–FockspaceHusimifunctionFockrearrangementsSchattennormsFaber–Krahn
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper proves that if you fix the area of a measurable set in the complex plane and look at the Toeplitz operator that multiplies by the indicator of that set and projects back onto the Fock space of entire functions, then the sum of its largest N eigenvalues is biggest precisely when the set is a disc. For N=1 that is the known Faber–Krahn inequality for the short-time Fourier transform; for general N it was a conjecture, previously settled only for radial sets. The argument rewrites the partial-sum objective as the maximal concentration of a Husimi function over rank-N projections, then uses a majorization theorem for density operators together with a bathtub comparison to force the optimizer to be a disc. As a direct consequence, every Schatten p-sum of the same operator is likewise optimized by discs, for the full range 0

What carries the argument

Ky Fan representation of the partial-sum objective as the maximal integral of the Husimi function Q_{ρ_P} of a normalized rank-N projection, compared via De Palma’s Husimi majorization (Fock/passive rearrangement of density operators) and the bathtub principle against the radial passive state built from the first N Fock basis vectors.

What would settle it

Exhibit any measurable set Ω of area s whose sum of the first N Toeplitz eigenvalues on the Fock space strictly exceeds the explicit disc value N−e^{−s}∑_{k=0}^{N−1}(N−k)s^k/k!.

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Extended reading notes

Core claim

For every measurable Ω⊂ℂ with finite area s and every integer N≥1, the Ky Fan sum of the first N eigenvalues of the Toeplitz operator T_Ω on the Fock space satisfies ∑_{j=1}^N λ_j(Ω) ≤ ∑_{j=1}^N λ_j(D_s), with the explicit right-hand side N−e^{−s}∑_{k=0}^{N−1}(N−k)s^k/k!, and equality for every disc of area s.

Load-bearing premise

The comparison rests on a majorization theorem from quantum information saying that, among density operators with a fixed spectrum, the one diagonal in the Fock basis with decreasing eigenvalues has the most concentrated Husimi function; if that fails for rank-N projections in this normalization, the disc comparison collapses.

Editorial extensions

If this is right

  • The N=1 case recovers the Faber–Krahn inequality for the short-time Fourier transform with Gaussian window.
  • All Schatten p-norms of T_Ω for 1<p≤∞ are maximized by discs of the same area, with explicit constants from the incomplete-gamma eigenvalues of the disc.
  • For 0<p<1 the corresponding Schatten quasi-norms (when finite) are minimized by the same discs.
  • The strong majorization λ(T_Ω)≺λ(T_{D_s}) holds for every finite-area set, so every symmetric convex functional of the eigenvalue sequence is optimized by discs.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same Ky-Fan-plus-Husimi route may be tryable for other coherent-state spaces where an analogous passive-rearrangement majorization is known, even if the classical differential-inequality methods fail.
  • Because individual eigenvalues λ_k for k>1 are not disc-maximized, the result isolates a genuine averaging phenomenon: only the ordered partial sums, not the separate modes, restore circular symmetry.
  • Uniqueness of maximizers up to null sets and translations is left open; a stability version of the majorization step would be a natural next quantitative question.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

0 major / 4 minor

Summary. The paper proves that among all measurable sets Ω⊂ℂ of finite area s, the disc D_s maximizes the Ky Fan norm σ_N(T_Ω)=∑_{j=1}^N λ_j(Ω) of the Toeplitz operator with symbol 1_Ω on the Fock space. Equality holds for every disc of area s. The N=1 case recovers Nicola–Tilli’s Faber–Krahn inequality; the general-N statement confirms the conjecture of Nicola–Riccardi–Tilli (previously known only for radial sets). The argument proceeds by identifying the Ky Fan sum with the maximal Husimi concentration of rank-N projections (Prop. 2.1 via Ky Fan’s principle), applying De Palma’s majorization theorem to the associated density operators after C^{1} approximation of (x−τ/N)_+ (Prop. 3.2), and comparing via the bathtub principle against the explicitly radial decreasing Husimi function of the passive state ρ_N. As a corollary, all Schatten p-sums (0<p≤∞) are likewise optimized by discs, via the resulting strong majorization of eigenvalue sequences and the infinite-dimensional Karamata principle.

Significance. The result settles a natural and explicitly posed conjecture that extends the Nicola–Tilli Faber–Krahn theorem from the operator norm to all finite Ky Fan norms, and simultaneously yields the full range of Schatten isoperimetric inequalities (previously known only for p=2 or under radial symmetry). The proof is short, modular, and conceptually novel: it imports De Palma’s Fock-rearrangement majorization from quantum information theory and combines it with classical tools (Ky Fan, bathtub). No smoothness or connectedness assumptions are imposed on Ω. Uniqueness up to null sets and translations is left open, which is appropriately acknowledged. The work therefore constitutes a clean, high-value contribution to the spectral geometry of localization operators on the Fock space.

minor comments (4)
  1. [§4] In the display after (4.1) and in the subsequent bathtub application, the threshold is written τ_s = Q_{ρ_N}(r_s); it would help the reader to record explicitly that this is the unique value satisfying |{Q_{ρ_N}>τ_s}|=s, so that the open disc is precisely the strict superlevel set.
  2. [Prop. 3.2] The approximation argument in Prop. 3.2 is correct (dominated convergence with 0≤Φ_ε≤x), but a one-sentence reference to the standard mollification of the positive-part function (or to the fact that De Palma’s theorem extends to continuous convex Φ by the same limit) would make the passage fully self-contained.
  3. [Title / front matter] Typographical inconsistencies appear in the title (“KY F AN”) and in several author-name accents (e.g., “LU´IS”); these should be normalized in the final version.
  4. [References] The arXiv identifiers of the two Nicola–Riccardi–Tilli preprints ([31], [32]) are given; once they appear, the published references should be updated.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Theorem 1 is assembled from independent external inputs (Ky Fan, De Palma majorization, bathtub, classical disc spectrum), none of which is defined in terms of the claimed maximizer.

full rationale

The derivation chain is modular and non-circular. Prop. 2.1 rewrites the Ky Fan sum via the classical Ky Fan maximum principle as a max of Husimi concentrations over rank-N projections. Prop. 3.2 applies De Palma’s external Husimi majorization theorem (after a standard C¹ approximation of the positive-part map) to compare any such concentration against the passive Fock rearrangement ρ_N. The bathtub lemma then forces the comparison set to be the centered disc D_s. Explicit diagonalization of T_{D_s} in the monomial basis identifies the right-hand side with ∑ λ_j(D_s). None of these ingredients assumes, fits, or is defined in terms of the isoperimetric claim being proved. Self-citations ([1]–[5]) appear only as historical background on concentration problems; the load-bearing citations are Ky Fan, De Palma–Trevisan–Giovannetti, and Nicola–Tilli / Nicola–Riccardi–Tilli (external authors). Uniqueness is explicitly left open and is not used. Corollary 1 is a routine majorization/Karamata consequence of Theorem 1 plus Tr(T_Ω)=|Ω|. Score 0.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The argument is a pure-math derivation with no fitted constants. It rests on standard operator-theoretic and rearrangement facts plus one deep external majorization theorem from quantum information. No new physical entities or free parameters are introduced.

assumptions (6)
  • standard math Ky Fan maximum principle: for a positive compact operator A, ∑_{j=1}^N λ_j(A) = max_{rank-N projections P} Tr(P A).
    Invoked as Prop. 2.1 / [13, Thm. 1] and [33, Thm. 11.13] to rewrite the Ky Fan norm as a maximal Husimi integral over rank-N projections.
  • domain assumption De Palma’s Husimi majorization: for density operators, ∫ Φ(Q_ρ) ≤ ∫ Φ(Q_{ρ↓}) for convex C¹ Φ with Φ(0)=0, where ρ↓ is the passive Fock rearrangement.
    Theorem 3.1 / [9, Thm. 5]; the load-bearing comparison tool. Itself rests on operator Lieb inequalities and passive-state optimality [10].
  • standard math Bathtub principle on ℂ: among sets of fixed measure, a superlevel set of an integrable nonnegative function maximizes the integral of that function.
    Lemma in §4, adapted from [1, Lem. 2.3] and [25]; applied to the radial decreasing profile Q_{ρ_N}.
  • domain assumption Fock-space Toeplitz operator T_Ω with symbol 1_Ω is positive, compact, trace-class when |Ω|<∞, with 0≤T_Ω≤I and Tr(T_Ω)=|Ω|.
    Standard facts recalled in §1 from the Daubechies/Seip localization-operator literature; used throughout.
  • standard math Centered discs diagonalize in the Fock monomial basis, with eigenvalues λ_{n+1}(D_s)=1−e^{-s}∑_{k=0}^n s^k/k! strictly decreasing in n.
    Computed in §4 via polar coordinates; classical (Seip, Daubechies). Identifies the sharp constant and confirms the first N eigenvalues are exactly those of e_0,…,e_{N-1}.
  • standard math Infinite-dimensional Karamata/majorization principle for convex and concave power maps on eigenvalue sequences.
    Cited from [31, Cor. 3.4, Prop. 3.5] to pass from Ky Fan majorization λ(T_Ω)≺λ(T_{D_s}) to all Schatten p-sums in Corollary 1.

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Pith. "Pith review of The isoperimetric inequality for the Ky Fan norm." pith.science (2026). https://pith.science/paper/64GYK3QW

@misc{pith2026260728376,
  author       = {Pith},
  title        = {Pith review of: The isoperimetric inequality for the Ky Fan norm},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/64GYK3QW}},
  note         = {Machine review of arXiv:2607.28376}
}
abstract

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 $.

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