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arxiv: 1907.10869 · v1 · pith:QO3JDDSWnew · submitted 2019-07-25 · 🧮 math.MG · math.AP· math.FA

Indecomposable sets of finite perimeter in doubling metric measure spaces

Paolo Bonicatto , Enrico Pasqualetto , Tapio Rajala This is my paper

Pith reviewed 2026-05-24 16:03 UTC · model grok-4.3

classification 🧮 math.MG math.APmath.FA
keywords indecomposable setsfinite perimeterdoubling metric measure spacesPoincaré inequalityBV functionsisotropicitymetric measure spaces
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The pith

Sets of finite perimeter in doubling metric measure spaces decompose into indecomposable components under an isotropicity assumption.

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

The paper develops a measure-theoretic notion of connectedness for sets of finite perimeter. It proves a decomposition theorem that splits any such set into indecomposable components and characterizes the extreme points of the space of BV functions. Both results are shown in doubling metric measure spaces that support a weak (1,1)-Poincaré inequality, but they require the additional isotropicity assumption on the space. This extends classical Euclidean results to general metric settings where geometry is controlled by doubling and Poincaré conditions rather than smoothness.

Core claim

In doubling metric measure spaces supporting a weak (1,1)-Poincaré inequality that also satisfy the isotropicity condition on the perimeter measure, every set of finite perimeter admits a decomposition into indecomposable sets, and the extreme points in the space of BV functions are characterized accordingly.

What carries the argument

The decomposition theorem into indecomposable sets, where indecomposability is the measure-theoretic inability to split a set into two subsets each carrying positive perimeter.

If this is right

  • Any set of finite perimeter decomposes into a countable union of indecomposable components.
  • The extreme points of the BV space admit a characterization in terms of indecomposable sets.
  • The results apply in any space meeting the doubling, Poincaré, and isotropicity conditions.

Where Pith is reading between the lines

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

  • The decomposition may fail in doubling Poincaré spaces that lack the isotropicity condition.
  • The results could supply new tools for minimal surfaces or isoperimetric problems in abstract metric spaces that satisfy the assumptions.
  • The extreme-point characterization might clarify the structure of the BV unit ball beyond Euclidean settings.

Load-bearing premise

The metric measure space satisfies the isotropicity assumption on the Hausdorff-type representation of the perimeter measure.

What would settle it

Construct or identify a doubling metric measure space with a weak (1,1)-Poincaré inequality that fails isotropicity, and find a set of finite perimeter that cannot be decomposed into indecomposable sets.

read the original abstract

We study a measure-theoretic notion of connectedness for sets of finite perimeter in the setting of doubling metric measure spaces supporting a weak $(1,1)$-Poincar\'{e} inequality. The two main results we obtain are a decomposition theorem into indecomposable sets and a characterisation of extreme points in the space of BV functions. In both cases, the proof we propose requires an additional assumption on the space, which is called isotropicity and concerns the Hausdorff-type representation of the perimeter measure.

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

1 major / 0 minor

Summary. The paper studies a measure-theoretic notion of connectedness for sets of finite perimeter in doubling metric measure spaces supporting a weak (1,1)-Poincaré inequality. It proves two main results under an additional isotropicity assumption on the Hausdorff-type representation of the perimeter measure: a decomposition theorem into indecomposable sets, and a characterization of extreme points in the space of BV functions.

Significance. If the results hold, they extend classical decomposition and extremal characterizations from Euclidean BV theory to a broader class of metric measure spaces. The decomposition theorem in particular supplies a structural tool that could support further work on connectedness and regularity questions in geometric measure theory on metric spaces.

major comments (1)
  1. [Abstract] Abstract: Both the decomposition theorem and the extremal characterization are stated to require the isotropicity assumption on the perimeter measure. This assumption lies outside the standard doubling + weak Poincaré setup and is load-bearing for the central claims; the manuscript should therefore quantify its restrictiveness (e.g., by exhibiting natural classes of spaces where it holds or fails) and clarify whether the results can be obtained without it.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the detailed reading and the constructive suggestion regarding the isotropicity assumption. We address the point below and will revise the manuscript accordingly to better quantify the assumption's scope.

read point-by-point responses

  1. Referee: [Abstract] Abstract: Both the decomposition theorem and the extremal characterization are stated to require the isotropicity assumption on the perimeter measure. This assumption lies outside the standard doubling + weak Poincaré setup and is load-bearing for the central claims; the manuscript should therefore quantify its restrictiveness (e.g., by exhibiting natural classes of spaces where it holds or fails) and clarify whether the results can be obtained without it.

    Authors: We agree that isotropicity is a non-standard assumption that is essential to both main theorems. In the revised version we will add a dedicated remark (likely after the statement of the main results) that explicitly lists classes of spaces in which isotropicity is known to hold, including Euclidean spaces, Carnot groups equipped with Haar measure, and Riemannian manifolds with the Riemannian volume measure. We will also indicate settings where the assumption is expected to fail, such as certain weighted Euclidean spaces with non-constant weights that destroy the isotropic representation of the perimeter measure. Our proofs make essential use of the isotropic representation of the perimeter measure to control the interaction between sets of finite perimeter; consequently we do not expect the stated decomposition or extremal characterization to hold in full generality without some form of this condition. The revision will make this dependence more transparent without claiming that weaker substitutes are currently available. revision: yes

Circularity Check

0 steps flagged

No significant circularity; theorems rest on explicit axioms plus isotropicity assumption

full rationale

The derivation chain consists of measure-theoretic theorems proved from doubling metric measure spaces supporting a weak (1,1)-Poincaré inequality, together with the explicitly stated isotropicity assumption on the perimeter measure. No load-bearing step reduces by construction to a fitted parameter, self-definition, or self-citation chain. The isotropicity hypothesis is introduced as an additional premise required for both main results, not derived from them. The paper is self-contained against its stated axioms.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the doubling and weak (1,1)-Poincaré properties of the ambient space plus the isotropicity condition introduced for the proofs; no free parameters or new entities are mentioned.

axioms (2)
  • domain assumption The ambient space is a doubling metric measure space supporting a weak (1,1)-Poincaré inequality.
    Explicitly stated as the setting in which the results hold.
  • ad hoc to paper The space satisfies the isotropicity assumption on the Hausdorff-type representation of the perimeter measure.
    Additional assumption required for both the decomposition and the extremal-point characterization.

pith-pipeline@v0.9.0 · 5610 in / 1359 out tokens · 22707 ms · 2026-05-24T16:03:56.023752+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

37 extracted references · 37 canonical work pages · 1 internal anchor

  1. [1]

    Ambrosio , Some Fine Properties of Sets of Finite Perimeter in Ahlfors R egular Metric Measure Spaces , Advances in Mathematics, 159 (2001), pp

    L. Ambrosio , Some Fine Properties of Sets of Finite Perimeter in Ahlfors R egular Metric Measure Spaces , Advances in Mathematics, 159 (2001), pp. 51–67

    work page 2001

  2. [2]

    , Fine Properties of Sets of Finite Perimeter in Doubling Metr ic Measure Spaces , Set-Valued Analysis, 10 (2002), pp. 111–128

    work page 2002

  3. [3]

    Ambrosio, E

    L. Ambrosio, E. Bru `e, and D. Semola , Rigidity of the 1-Bakry- ´Emery inequality and sets of finite perimeter in RCD spaces, (2018). Accepted at Geom. Funct. Anal., arXiv:1812.07890

    work page arXiv 2018

  4. [4]

    Ambrosio, V

    L. Ambrosio, V. Caselles, S. Masnou, and J.-M. Morel , Connected components of sets of finite perimeter and applications to image processing , Journal of the European Mathematical Society, 3 (2001), pp . 39–92

    work page 2001

  5. [5]

    Ambrosio and S

    L. Ambrosio and S. Di Marino , Equivalent definitions of BV space and of total variation on m etric measure spaces, Journal of Functional Analysis, 266 (2014), pp. 4150–4188

    work page 2014

  6. [6]

    Ambrosio, N

    L. Ambrosio, N. Fusco, and D. Pallara , Functions of Bounded Variation and Free Discontinuity Prob lems, Oxford Science Publications, Clarendon Press, 2000

    work page 2000

  7. [7]

    Ambrosio, M

    L. Ambrosio, M. Miranda, and D. Pallara , Special functions of bounded variation in doubling metric measure spaces, Quad. Mat., 14 (2004). INDECOMPOSABLE SETS OF FINITE PERIMETER IN DOUBLING METRIC MEASURE SPACES 31

    work page 2004

  8. [8]

    Baldi , Weighted BV functions , Houston Math

    A. Baldi , Weighted BV functions , Houston Math. J., 27 (2001)

    work page 2001

  9. [9]

    Bellettini, G

    G. Bellettini, G. Bouchitt ´e, and I. Fragal `a, BV functions with respect to a measure and relaxation of metric integral functionals , Journal of Convex Analysis, 6 (1999), pp. 349–366

    work page 1999

  10. [10]

    Biroli and U

    M. Biroli and U. Mosco , Sobolev and isoperimetric inequalities for Dirichlet form s on homogeneous spaces , Atti Accad. Naz. Lincei. Cl. Sci. Fis. Mat. Natur., 6 (1995), pp. 37–44

    work page 1995

  11. [11]

    V. I. Bogachev , Measure Theory, Measure Theory, Springer Berlin Heidelberg, 2007

    work page 2007

  12. [12]

    Bonicatto and N

    P. Bonicatto and N. A. Gusev , On the structure of divergence-free measures in R2, (2019). In preparation

    work page 2019

  13. [13]

    Caccioppoli , Misura e integrazione sugli insiemi dimensionalmente orie ntati, Accad

    R. Caccioppoli , Misura e integrazione sugli insiemi dimensionalmente orie ntati, Accad. Naz. Lincei, 12 (1952), pp. 3–11 and 137–146

    work page 1952

  14. [14]

    Danielli, N

    D. Danielli, N. Garofalo, and D.-M. Nhieu , Trace inequalities for Carnot-Carath´ eodory spaces and appli- cations, Ann. Scuola Norm. Sup., 27 (1998), pp. 195–252

    work page 1998

  15. [15]

    De Giorgi , Su una teoria generale della misura (r − 1)-dimensionale in uno spazio ad r dimensioni, Annali di Matematica Pura ed Applicata, 36 (1954), pp

    E. De Giorgi , Su una teoria generale della misura (r − 1)-dimensionale in uno spazio ad r dimensioni, Annali di Matematica Pura ed Applicata, 36 (1954), pp. 191–213

    work page 1954

  16. [16]

    Di Marino , Recent advances on BV and Sobolev spaces in metric measure sp aces, (2014)

    S. Di Marino , Recent advances on BV and Sobolev spaces in metric measure sp aces, (2014). PhD Thesis

    work page 2014

  17. [17]

    Sobolev and BV spaces on metric measure spaces via derivations and integration by parts

    , Sobolev and BV spaces on metric measure spaces via derivatio ns and integration by parts , (2014). Submitted paper, arXiv:1409.5620

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  18. [18]

    Diestel and J

    J. Diestel and J. J. Uhl , Vector Measures, Mathematical surveys and monographs, American Mathemati cal Society, 1977

    work page 1977

  19. [19]

    Dolzmann and S

    G. Dolzmann and S. M ¨uller, Microstructures with finite surface energy: the two-well pr oblem, Archive for Rational Mechanics and Analysis, 132 (1995), pp. 101–141

    work page 1995

  20. [20]

    Federer , Geometric measure theory , Grundlehren der mathematischen Wissenschaften, Springe r, 1969

    H. Federer , Geometric measure theory , Grundlehren der mathematischen Wissenschaften, Springe r, 1969

    work page 1969

  21. [21]

    W. H. Fleming , Functions with generalized gradient and generalized surfa ces, Annali di Matematica Pura ed Applicata, 44 (1957), pp. 93–103

    work page 1957

  22. [22]

    Math., 4 (1960), pp

    , Functions whose partial derivatives are measures , Illinois J. Math., 4 (1960), pp. 452–478

    work page 1960

  23. [23]

    Franchi, R

    B. Franchi, R. Serapioni, and S. C. Francesco , Rectifiability and perimeter in the Heisenberg group , Mathematische Annalen, 321 (2001), pp. 479–531

    work page 2001

  24. [24]

    Franchi, R

    B. Franchi, R. Serapioni, and F. Serra Cassano , Meyers–Serrin type theorems and relaxation of variational integrals depending on vector fields , Houston Math. J., 22 (1996), pp. 859–889

    work page 1996

  25. [25]

    , On the structure of finite perimeter sets in step 2 Carnot grou ps, The Journal of Geometric Analysis, 13 (2003), p. 421

    work page 2003

  26. [26]

    Garofalo and D.-M

    N. Garofalo and D.-M. Nhieu , Isoperimetric and Sobolev inequalities for Carnot-Carath ´ eodory spaces and the existence of minimal surfaces , Comm. Pure Appl. Math., 49 (1996), pp. 1081–1144

    work page 1996

  27. [27]

    Giusti , Minimal surfaces and functions of bounded variation , Birkh¨ auser, 1985

    E. Giusti , Minimal surfaces and functions of bounded variation , Birkh¨ auser, 1985

    work page 1985

  28. [28]

    Heinonen and P

    J. Heinonen and P. Koskela , Quasiconformal maps in metric spaces with controlled geome try, Acta Math., 181 (1998), pp. 1–61

    work page 1998

  29. [29]

    Heinonen, P

    J. Heinonen, P. Koskela, N. Shanmugalingam, and J. T. Tyson , Sobolev spaces on metric measure spaces: An approach based on upper gradients , vol. 27 of New Mathematical Monographs, Cambridge Univers ity Press, 2015

    work page 2015

  30. [30]

    R. A. Johnson , Atomic and Nonatomic Measures , Proceedings of the American Mathematical Society, 25 (1970), pp. 650–655

    work page 1970

  31. [31]

    Kirchheim , Lipschitz minimizers of the 3-well problem having gradient s of bounded variation , (1998)

    B. Kirchheim , Lipschitz minimizers of the 3-well problem having gradient s of bounded variation , (1998). Preprint 12, Max Planck Institute for Mathematics in the Sci ences, Leipzig

    work page 1998

  32. [32]

    Krein and D

    M. Krein and D. Milman , On extreme points of regular convex sets , Studia Math., 9 (1940), pp. 133–138

    work page 1940

  33. [33]

    J. M. Mackay, J. T. Tyson, and K. Wildrick , Modulus and Poincar´ e Inequalities on Non-Self-Similar Sierpi´ nski Carpets, Geometric and Functional Analysis, 23 (2013), pp. 985–103 4

    work page 2013

  34. [34]

    Mattila , Geometry of Sets and Measures in Euclidean Spaces: Fractals and Rectifiability , Cambridge Studies in Advanced Mathematics, Cambridge University Pre ss, 1995

    P. Mattila , Geometry of Sets and Measures in Euclidean Spaces: Fractals and Rectifiability , Cambridge Studies in Advanced Mathematics, Cambridge University Pre ss, 1995

    work page 1995

  35. [35]

    Milman , Characteristics of extreme points of regularly convex sets , Dokl

    D. Milman , Characteristics of extreme points of regularly convex sets , Dokl. Akad. Nauk. SSSR, 57 (1947), pp. 119–122

    work page 1947

  36. [36]

    Miranda , Functions of bounded variation on “good” metric spaces , Journal de Math´ ematiques Pures et Appliqu´ ees, 82 (2003), pp

    M. Miranda , Functions of bounded variation on “good” metric spaces , Journal de Math´ ematiques Pures et Appliqu´ ees, 82 (2003), pp. 975–1004

    work page 2003

  37. [37]

    R. R. Phelps , Lectures on Choquet’s Theorem , Lecture Notes in Mathematics, Springer Berlin Heidelberg , 2003. 32 PAOLO BONICATTO, ENRICO PASQUALETTO, AND TAPIO RAJALA Departement Mathematik und Informatik, Universit¨at Basel, Spiegelgasse 1, CH-4051, Basel, Switzer- land E-mail address : paolo.bonicatto@unibas.ch University of Jyvaskyla, Department of M...

    work page 2003