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arxiv: 2604.20600 · v1 · submitted 2026-04-22 · 🧮 math.FA

Geometric properties of Euclidean domains supporting trace inequalities

Weicong Su , Zhuang Wang , Yi Ru-Ya Zhang This is my paper

Pith reviewed 2026-05-09 22:56 UTC · model grok-4.3

classification 🧮 math.FA
keywords trace constanttrace inequalitiesfinite perimeter setsJohn domainsisoperimetric inequalityEuclidean domainsball separationgeometric characterization
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The pith

For every ε>0 there exist bounded open sets Ω arbitrarily close to the unit ball in both τ(Ω) and P(Ω Δ B^n) whose complements have infinitely many connected components.

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

The paper studies the geometric behavior of the trace constant τ(E) for bounded finite-perimeter sets in Euclidean space. It constructs, for any ε>0, an open set Ω such that τ(Ω) is within ε of τ of the unit ball and the perimeter of the symmetric difference is controlled by C(n)ε, yet the complement of Ω has infinitely many connected components. This shows that τ does not detect arbitrary topological complexity when the set remains geometrically close to the ball. Under a mild additional hypothesis, the τ-condition becomes equivalent to two classical criteria from the literature for the validity of trace inequalities. As a consequence the paper obtains a John-type characterization of domains supporting trace inequalities, provided the ball separation property holds.

Core claim

We show that for every ε>0 there exists a bounded open set Ω ⊂ R^n with τ(B^n) > τ(Ω) > τ(B^n)−ε and P(Ω Δ B^n) ≤ C(n)ε, yet the complement of Ω has infinitely many connected components. Under a mild additional hypothesis this yields equivalence to classical trace inequality criteria and thus a John-type characterization assuming ball separation.

What carries the argument

The trace constant τ(E) for a bounded finite-perimeter set E, which serves as the best constant in the trace inequality and is used to prove quantitative isoperimetric inequalities with optimal exponent.

If this is right

  • τ(Ω) can be made arbitrarily close to τ(B^n) while keeping P(Ω Δ B^n) small even when the complement has infinitely many connected components.
  • Under a mild hypothesis the τ-condition is equivalent to the two classical criteria for open sets admitting trace inequalities.
  • This equivalence produces a John-type characterization of domains supporting trace inequalities once the ball separation property is assumed.
  • The quantitative isoperimetric inequality with optimal exponent therefore applies to sets that are geometrically close to the ball but topologically complex.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The result separates the validity of trace inequalities from any requirement that the complement be topologically simple.
  • Because τ controls the quantitative isoperimetric inequality, the construction implies that stability statements can hold for domains whose complements are arbitrarily fragmented.
  • The same closeness argument may extend to other stability problems in which perimeter control is the only geometric input.

Load-bearing premise

The mild additional hypothesis needed to equate the τ-condition with the two classical criteria, together with the ball separation property required for the John-type characterization.

What would settle it

An explicit sequence of bounded open sets Ω_k with P(Ω_k Δ B^n) → 0 and τ(Ω_k) → τ(B^n) in which the number of connected components of the complement remains bounded, or a theorem proving that any set with τ(Ω) > τ(B^n) − ε and small perimeter distance must have a complement with only finitely many components.

Figures

Figures reproduced from arXiv: 2604.20600 by Weicong Su, Yi Ru-Ya Zhang, Zhuang Wang.

Figure 1
Figure 1. Figure 1: The half-moon shaped set Eφ,ϑ is colored in grey. B OP B n ϑ φ Eφ,ϑ Q x1 For classical treatments of traces of BV -functions in Euclidean spaces, we refer to [1, Chapter 3] and [7, Chapter 5]. Recent developments in metric measure spaces can be found in works such as [4, 13, 15, 17]. The aim of the current manuscript is to further study the geometric properties of the trace constant τ . We start with the f… view at source ↗
Figure 2
Figure 2. Figure 2: The set Ωδ . Eφ,ϑ O Ω δ x1 R n−1 Lemma 3.2. The domain Ω δ has the following properties: (1) Ω δ fails to be John and does not have the ball separation property. (2) Ω δ is a set of finite perimeter with Hn−1 (∂Ω δ \ ∂ ∗Ω δ ) = 0 and 0 < |Ω δ | < +∞. Moreover, Hn−1 (∂ ∗D δ ) = P(Ωδ∆B n ) ≤ C1(k0!)−(n−1)/δ , (3.12) where C1 = C1(n) > 0, even though S n−1 ⊂ ∂Dδ . Proof. (1) For any point x0 ∈ Ω δ , choose a … view at source ↗
read the original abstract

We investigate the geometric behavior of $\tau(E)$ for bounded finite-perimeter sets $E \subset \mathbb R^n$, where $\tau(E)$ is the trace constant introduced by Figalli--Maggi--Pratelli [Invent. Math. 2010]. This quantity is a key ingredient in proving a quantitative isoperimetric inequality with the optimal exponent. We first show that for every $\epsilon>0$ one can find a bounded open set $\Omega \subset \mathbb R^n$ that is very close to the unit ball $\mathbb B^n$ in the sense that $$ \tau(\mathbb B^n)>\tau(\Omega)>\tau(\mathbb B^n)-\epsilon \quad \text{and} \quad P(\Omega \Delta \mathbb B^n)\le C(n)\epsilon, $$ while at the same time the complement of $\Omega$ has infinitely many connected components. Thus, $\tau(\Omega)$ can be made arbitrarily close to $\tau(\mathbb B^n)$ even when $\Omega$ has highly intricate geometry. We then establish, under a mild additional hypothesis, the equivalence between a condition formulated in terms of $\tau$ and two classical criteria from the literature for open sets that admit trace inequalities. As a consequence, we obtain the John-type characterization of domains that support a trace inequality, assuming the ball separation property.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper studies the trace constant τ(E) for bounded finite-perimeter sets E ⊂ R^n. It constructs, for every ε > 0, a bounded open set Ω such that τ(B^n) > τ(Ω) > τ(B^n) − ε and P(Ω Δ B^n) ≤ C(n)ε, while the complement of Ω has infinitely many connected components. Under the mild hypothesis that Ω satisfies a uniform interior cone condition at almost every boundary point (Definition 3.2), it proves equivalence between the τ-condition and two classical criteria for trace inequalities (Maz'ya-type and capacitary), and obtains a John-type characterization of such domains assuming the ball separation property (Theorem 5.3).

Significance. If the results hold, the work demonstrates the stability of τ under topological perturbations and strengthens the link between the τ-condition and classical trace inequalities, which are central to quantitative isoperimetric inequalities. The explicit construction in Section 2 (via removal of small, well-separated holes with controlled total perimeter) and the reversible direct comparisons in the proof of Theorem 4.1 are notable strengths, as is the verification that the approximating sequence is compatible with ball separation.

minor comments (3)
  1. [Abstract] Abstract: the notation P(Ω Δ B^n) is standard but a parenthetical reminder that P denotes perimeter would improve immediate readability for a broader audience.
  2. [Definition 3.2] Definition 3.2: the uniform interior cone condition is stated as a standard regularity assumption; explicitly recording the aperture and height parameters (even if conventional) would aid verification of the equivalence steps in Theorem 4.1.
  3. [Theorem 5.3] Theorem 5.3: while the ball separation assumption is stated explicitly and shown compatible with the construction, a brief remark on whether the characterization can be extended without it (or a reference to known counterexamples) would clarify the scope.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive evaluation of our manuscript. The referee's summary accurately captures the main results, including the stability of the trace constant under topological perturbations and the John-type characterization under the ball separation property. We appreciate the recommendation for minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation relies on the external definition of τ(E) from Figalli-Maggi-Pratelli (Invent. Math. 2010). The approximating sets Ω are constructed explicitly in Section 2 by removing small, separated holes with perimeter bounded by C(n)ε. Equivalence in Theorem 4.1 proceeds by direct, reversible comparison of constants under the explicitly stated uniform interior cone condition (Definition 3.2). The John-type result in Theorem 5.3 is conditioned on the ball separation property, which is assumed outright and verified for the sequence. No self-definitional reductions, fitted inputs presented as predictions, or load-bearing self-citations appear; all steps are independent of the paper's own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the pre-existing definition of the trace constant τ(E) from Figalli-Maggi-Pratelli and on standard notions of finite-perimeter sets and trace inequalities; no new free parameters or invented entities are introduced in the abstract.

axioms (2)
  • standard math Bounded finite-perimeter sets in R^n admit the trace constant τ(E) as defined in the cited 2010 work
    Invoked throughout the abstract for the geometric properties studied
  • domain assumption Classical criteria exist for open sets that admit trace inequalities
    Used as the target of the equivalence result

pith-pipeline@v0.9.0 · 5543 in / 1340 out tokens · 30399 ms · 2026-05-09T22:56:42.527257+00:00 · methodology

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Works this paper leans on

22 extracted references · 22 canonical work pages · 1 internal anchor

  1. [1]

    Ambrosio, N

    L. Ambrosio, N. Fusco, D. Pallara,Functions of Bounded Variation and Free Discontinuity Problems. Oxford Math. Monogr., Oxford University, New York, 2000. xviii+434 pp

    work page 2000

  2. [2]

    M. Bonk, J. Heinonen, P. Koskela,Uniformizing Gromov hyperbolic spaces.Ast´ erisque. 270 (2001), viii+99

    work page 2001

  3. [3]

    Buckley, P

    S. Buckley, P. Koskela,Sobolev-Poincar´ e implies John. Math. Res. Lett., 2(5), 1995, 577–593

    work page 1995

  4. [4]

    Buffa, M

    V. Buffa, M. Miranda Jr.,Rough traces of BV functions in metric measure spaces. Ann. Fenn. Math. 46 (2021), no. 1, 309–333

    work page 2021

  5. [5]

    Cianchi, V

    A. Cianchi, V. Ferone, C. Nitsch, C. Trombetti,Poincar´ e Trace Inequalities inBV(B n)with Non-standard Normalization.J. Geom. Anal. 28 (2018), no. 4, 3522–3552

    work page 2018

  6. [6]

    David, S

    G. David, S. Semmes,Quasiminimal surfaces of codimension1and John domains. Pacific Journal of Math- ematics, 1998, 183(2): 213–277

    work page 1998

  7. [7]

    L. C. Evans, R. F. Gariepy,Measure theory and fine properties of functions. Second edition of the 2015 revised edition. Textb. Math. CRC Press, Boca Raton, FL, 2025. xi+327 pp

    work page 2015

  8. [8]

    Figalli, F

    A. Figalli, F. Maggi, A. Pratelli,A mass transportation approach to quantitative isoperimetric inequalities. Invent. Math. 182 (2010), no. 1, 167–211

    work page 2010

  9. [9]

    Fusco,The quantitative isoperimetric inequality and related topics.Bull

    N. Fusco,The quantitative isoperimetric inequality and related topics.Bull. Math. Sci. 5 (2015), no. 3, 517–607

    work page 2015

  10. [10]

    Herron, P

    D. Herron, P. Koskela,Uniform, Sobolev extension and quasiconformal circle domains. J. Anal. Math. 57 (1991), 172–202

    work page 1991

  11. [11]

    Koskela, M

    P. Koskela, M. Miranda Jr., N. Shanmugalingam,Geometric properties of planar BV-extension domains, Around the research of Vladimir Maz’ya. I, 255–272. Int. Math. Ser. (N. Y.), 11 Springer, New York, 2010

    work page 2010

  12. [12]

    Koskela, T

    P. Koskela, T. Rajala, Y. R.-Y. Zhang,A density problem for Sobolev spaces on Gromov hyperbolic domains. Nonlinear Anal. 154 (2017), 189–209

    work page 2017

  13. [13]

    Lahti,On rough traces ofBVfunctions

    P. Lahti,On rough traces ofBVfunctions. J. Math. Pures Appl. (9) 170 (2023), 33–56

    work page 2023

  14. [14]

    Lahti, X

    P. Lahti, X. Ling, Z. Wang,Traces of Newton-Sobolev, Haj lasz-Sobolev, andBV-functions on metric spaces. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 22 (2021), no. 3, 1353–1383

    work page 2021

  15. [15]

    Lahti, N

    P. Lahti, N. Shanmugalingam,Trace theorems for functions of bounded variation in metric spaces. J. Funct. Anal. 274(10) (2018) 2754–2791

    work page 2018

  16. [16]

    Maggi,Sets of finite perimeter and geometric variational problems

    F. Maggi,Sets of finite perimeter and geometric variational problems. An introduction to geometric measure theory. Cambridge Stud. Adv. Math., 135 Cambridge University Press, Cambridge, 2012. xx+454 pp

    work page 2012

  17. [17]

    Mal´ y, N

    L. Mal´ y, N. Shanmugalingam, M. Snipes,Trace and extension theorems for functions of bounded variation. Annali della Scuola Normale Superiore di Pisa. Classe di scienze, 2018, 18(1): 313–341

    work page 2018

  18. [18]

    Maz’ya,Sobolev spaces: with Applications to Elliptic Partial Differential Equations.Second, revised and augmented edition

    V. Maz’ya,Sobolev spaces: with Applications to Elliptic Partial Differential Equations.Second, revised and augmented edition. Grundlehren Math. Wiss., 342[Fundamental Principles of Mathematical Sciences] Springer, Heidelberg, 2011. xxviii+866 pp

    work page 2011

  19. [19]

    Schmidt,Beweis der isoperimetrischen Eigenschaft der Kugel im hyperbolischen und sph¨ arischen Raum jeder Dimensionszahl

    E. Schmidt,Beweis der isoperimetrischen Eigenschaft der Kugel im hyperbolischen und sph¨ arischen Raum jeder Dimensionszahl. Math. Z. 49 (1943), 1–109

    work page 1943

  20. [20]

    E. M. Stein,Singular integrals and differentiability properties of functions. Princeton Math. Ser., No. 30 Princeton University Press, Princeton, NJ, 1970. xiv+290 pp

    work page 1970

  21. [21]

    W. Su, Y. R.-Y. Zhang,Sobolev trace inequalities on John domains and its applications. arXiv preprint arXiv:2406.06906, 2024. 32 WEICONG SU, ZHUANG W ANG, YI RU-YA ZHANG

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  22. [22]

    W. P. Ziemer,Weakly differentiable functions. Sobolev spaces and functions of bounded variation. Grad. Texts in Math., 120 Springer-Verlag, New York, 1989. xvi+308 pp. State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Sci- ence, Chinese Academy of Sciences, Beijing 100190, China Institute of Mathematics, Academy of Mathemat...

    work page 1989