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Absence of non-compactly supported minimisers for the Lieb-Oxford bound

T0 review · 0 major / 5 minor · reviewed 2026-07-14 · grok-4.5

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Pith's one-line read Minimisers of the fixed-N Lieb–Oxford bound, if they exist, must be compactly supported.

desk verdict Clean extension of Lieb–Oxford compactness from N=1 to all finite N; proof holds and existence is correctly left open. read the letter →

arxiv 2607.11440 v1 pith:56EYKO3Z submitted 2026-07-13 math.AP math-phmath.MP

Simone Di Marino , Rodrigue Lelotte This is my paper · ORCID
classification math.APmath-phmath.MP MSC 49Q2235Q4049J45
keywords Lieb-OxfordinequalitymultimarginaloptimaltransportKantorovichpotentialRieszcostcompactsupportfixedparticlenumber
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

The Lieb–Oxford inequality supplies a universal lower bound on the indirect Riesz energy of a system of charged particles. For a fixed finite number of particles N the paper asks whether a density that attains the best possible constant can fail to have compact support. The answer is no: any minimiser must be compactly supported. The argument proceeds by treating the indirect energy as a multimarginal optimal-transport problem, extracting a Kantorovich potential from the first-order optimality condition, and comparing its known decay at infinity with the Riesz potential generated by the density. The resulting contradiction forces the support to be bounded. The result extends the classical one-particle observation of Lieb and Oxford and supplies a structural constraint that any future existence proof or numerical search for the sharp constant must respect.

What carries the argument

The asymptotic expansion of c-conjugated Kantorovich potentials: any such potential associated with a density of unbounded support behaves at infinity like (N-1)/|x|^s plus a constant. Combined with the variational characterisation that identifies U_ρ - eta_s c_LO ho^{s/d} as a Kantorovich potential, this expansion produces incompatible decay rates once the support is assumed unbounded.

What would settle it

Exhibit a density of total mass N that is unbounded in support, belongs to L^{1} igcap L^{1+s/d}, and attains the sharp constant c_LO(s,d,N); alternatively, produce a c-conjugated Kantorovich potential whose far-field expansion violates the (N-1)/|x|^s law on an unbounded support.

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Extended reading notes

Core claim

If a density ρ of total mass N attains the sharp constant c_LO(s,d,N) in the fixed-N Lieb–Oxford inequality, then ρ is necessarily compactly supported. The claim is proved for every dimension d and every Riesz exponent 0 < s < d, and it does not assert existence of such a minimiser.

Load-bearing premise

The proof relies on the known far-field expansion of every c-conjugated Kantorovich potential for densities of unbounded support; if that expansion fails for some admissible density the contradiction argument collapses.

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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 / 5 minor

Summary. The paper proves that any minimiser (if it exists) of the fixed-N Lieb–Oxford constant c_LO(s,d,N) must be compactly supported. For ρ in L^{1} ∩ L^{1+s/d} with mass N that attains the bound, the authors derive a variational inequality showing that ϕ_ρ := U_ρ − β_s c_LO ρ^{s/d} is a Kantorovich potential for the multimarginal Riesz OT problem. After replacing it by a c-conjugated Lipschitz representative ϕ (via a strong duality formulation (KD’) proved in the appendix), they obtain inner/outer estimates relating U_ρ, ϕ and ρ. A bootstrap that combines these estimates with the Hardy–Littlewood–Sobolev inequality upgrades local integrability of ρ to global L^q integrability for some q > d/(d−s), which forces U_ρ → 0 at infinity and identifies the additive constant of ϕ with its limit at infinity. Assuming unbounded support then yields a two-sided contradiction: the complement of the support cannot be unbounded (by the asymptotic expansion of ϕ), while a bounded complement forces a non-integrable tail of ρ. The argument extends the classical N=1 result of Lieb–Oxford and leaves existence of minimisers open.

Significance. The result settles a natural structural question for the fixed-N Lieb–Oxford problem that had been known only for N=1 and only heuristically for N>1. Compact support of putative minimisers is a useful a-priori restriction for any future existence or numerical search, and the proof technique (variational characterisation of Kantorovich potentials + HLS bootstrap + asymptotic comparison) is clean and reusable. The appendix supplies a self-contained justification of the essential-infimum duality (KD’) that is needed because the candidate potential arises only almost everywhere. The paper correctly treats existence as open and relies on a recent, carefully cited asymptotic lemma for c-conjugated potentials; under those hypotheses the argument is complete.

minor comments (5)
  1. Page 1 and abstract: the arXiv identifier and date appear as 2607.11440 / July 14, 2026; these look like placeholders and should be corrected before publication.
  2. Section 2, after (KD’): the set I_ρ is defined as L^{1}_loc ∩ L^{1}(ρ); a short remark that this is the natural space for the variational inequality (5) would help the reader who is used to continuous dual potentials.
  3. Lemma 3 and the subsequent global-integrability bootstrap: the iteration q ↦ sq/(d−(d−s)q) is clear, but an explicit sentence that the same map works globally once ϕ−c ≥ 0 would make the transition from local to global more transparent.
  4. Appendix A.1: the successive construction of the ϕ_k is correct but a little dense; a one-line indication that each step only uses Fubini and the definition of essential infimum would improve readability.
  5. References: a few entries (e.g. [7], [8]) are listed as “to appear” or with incomplete pagination; final bibliographic data should be supplied if available.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: pure contradiction proof that never feeds the target statement back into its hypotheses.

full rationale

Theorem 1 is proved by reductio: assume an LO minimiser ρ of unbounded support, extract a Kantorovich potential φ_ρ from the first-order variational inequality (17) via Lemma 1, upgrade it by Theorem 2 to a c-conjugated Lipschitz KP φ, bootstrap local-to-global integrability of ρ by HLS + the inner estimate (20), obtain c = C_φ, then derive contradictory asymptotics both outside and inside the support via Lemma 2. None of these steps defines the conclusion into the premises. The only external inputs are standard duality results and the asymptotic expansion of c-conjugated KPs (Lemma 2, cited from Lelotte 2025); that expansion is an independent theorem about the multimarginal OT problem and does not presuppose compactness of LO minimisers. Existence of minimisers is left open, so there is no self-referential existence claim. Score 0 is therefore the correct assessment.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters or invented physical entities. The argument rests on standard functional-analytic facts (HLS, Lebesgue differentiation, convex duality) plus two domain-specific results on multimarginal OT with Riesz costs that are cited and used as black boxes.

assumptions (4)
  • standard math Hardy–Littlewood–Sobolev inequality relating L^{1+s/d} densities to their Riesz potentials
    Used repeatedly in the regularity bootstrap of Lemma 3 and the global integrability argument that forces U_ ho o0 at infinity.
  • domain assumption Existence of a c-conjugated, Lipschitz, uniformly bounded Kantorovich potential for the dual (KD’) when ho is absolutely continuous of mass N
    Taken from Buttazzo–Champion–De Pascale and Colombo–Di Marino–Stra; invoked as Theorem 2 and used to obtain the inner/outer estimates (19)–(20).
  • domain assumption Asymptotic expansion φ(x)=(N-1)/|x|^s + C_φ + o(1/|x|^s) for c-conjugated Kantorovich potentials when supp ho is unbounded
    Lemma 2, proved in Lelotte 2025; the contradiction in the final paragraph of Section 3 relies on the precise coefficient N-1.
  • ad hoc to paper Strong Kantorovich duality (KD’) holds with essential-infimum constraint
    Proved in Appendix A; needed because the candidate potential φ_ ho arises only almost everywhere from the first variation.

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Cite this review

Pith. "Pith review of Absence of non-compactly supported minimisers for the Lieb-Oxford bound." pith.science (2026). https://pith.science/paper/56EYKO3Z

@misc{pith2026260711440,
  author       = {Pith},
  title        = {Pith review of: Absence of non-compactly supported minimisers for the Lieb-Oxford bound},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/56EYKO3Z}},
  note         = {Machine review of arXiv:2607.11440}
}
abstract

We prove the minimisers of the Lieb-Oxford bound (if any) for a fixed (and finite) number of particles $N \geq 1$ are necessarily compactly supported, extending a result proved by E. H. Lieb and S. Oxford in the one-particle case.

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Reference graph

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