Pith. sign in

REVIEW 1 major objections 5 minor 37 references

Uniqueness of critical points of the anisotropic isoperimetric problem for finite perimeter sets

T0 review · 1 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Under a mild boundary-closure hypothesis, every finite-perimeter set with anisotropic mean curvature in $(0,c]$ satisfies a sharp Heintze-Karcher inequality, and equality forces finite unions of disjoint open Wulff shapes of radius at…

desk verdict A strong anisotropic Heintze-Karcher theorem, but Corollary 6.8 has a real gap in its proof that looks repairable. read the letter →

arxiv 1908.09795 v1 pith:VSKS6GTB submitted 2019-08-26 math.AP math.DG

Antonio De Rosa , S{l}awomir Kolasiński , Mario Santilli This is my paper · ORCID
classification math.APmath.DG MSC 49Q2049Q10
keywords anisotropicisoperimetricproblemWulffshapefiniteperimetersetsmeancurvatureHeintze-Karcherinequalityvarifoldscriticalpointsnormalbundle
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 paper attacks a rigidity conjecture for the anisotropic isoperimetric problem at the low regularity of finite-perimeter sets. It shows that, for a $\mathscr{C}^3$ elliptic integrand $F$, any finite-perimeter set $E$ whose reduced boundary is almost closed and whose anisotropic mean curvature lies between $0$ and $c$ satisfies the anisotropic Heintze-Karcher inequality $L^{n+1}(E) \le \frac{n}{n+1}\int_{\partial E} \frac{1}{|h_F(V,x)|}\,dH^n(x)$. If equality holds, $E$ must coincide, up to a null set, with a finite union of disjoint open Wulff shapes of radii at least $n/c$. Consequently, under the same boundary condition, volume-constrained critical points of the anisotropic perimeter are precisely finite unions of disjoint equal-radius Wulff shapes. The result extends the smooth-boundary rigidity theorem to finite-perimeter sets, up to the stated boundary condition.

What carries the argument

The load-bearing object is the generalized anisotropic unit normal bundle $N^F(A)=\{(a,\operatorname{grad}F(u)):(a,u)\in N(A)\}$ together with the anisotropic nearest-point projection $\xi_A^F$ and the anisotropic reach $r^F_A$. The proof establishes an $n$-dimensional Lusin (N) property for $N^F$ on anisotropic $(n,h)$-sets, which lets mean-curvature bounds on the boundary be transferred to the $\mathscr{C}^{1,1}$ level sets of the anisotropic distance function. A coarea and area integration of the Jacobian of the anisotropic normal map yields the volume estimate, and the equality case uses the Steiner-formula theorem to show the anisotropic reach is at least $n/c$, then a totally-umbilical classification: a connected $\mathscr{C}^{1,1}$ hypersurface all of whose $F$-principal curvatures are equal is the boundary of a Wulff shape.

What would settle it

Search for a compact smooth hypersurface $\Sigma\subset\mathbb{R}^3$ of genus one and a $\mathscr{C}^3$ elliptic integrand $F$ for which the anisotropic mean curvature of $\Sigma$ is a positive constant. If one exists, the set enclosed by $\Sigma$ satisfies the boundary regularity hypotheses but is not a Wulff shape, contradicting the equality case of the theorem.

Watch

Extended reading notes

Core claim

The central claim is Theorem 6.4. Let $F$ be an elliptic integrand of class $\mathscr{C}^3$, let $E\subseteq\mathbb{R}^{n+1}$ be a finite-perimeter set with $H^n(\mathrm{Clos}(\partial^*E)\setminus\partial^*E)=0$, and let $V=v_n(\partial^*E)$ be its boundary varifold. Suppose $V$ has no singular first variation, its anisotropic mean curvature satisfies $0<-h_F(V,x)\cdot n(E,x)\le c$, and $h_F(V,\cdot)$ is $\mathscr{C}^{0,\alpha}$ on compact subsets of the $\mathscr{C}^{1,\alpha}$ regular part of $\operatorname{spt}\|V\|$. Then the anisotropic Heintze-Karcher inequality holds. Equality holds exactly when $E$ differs by a set of measure zero from a finite union of disjoint open Wulff shapes of radii not smaller than $n/c$. Corollary 6.8 draws the critical-point conclusion: a finite-volume finite-perimeter set satisfying the same boundary-closure condition that is a volume-constrained critical point of $P_F$ must be a finite union of disjoint equal-radius Wulff shapes.

Load-bearing premise

The argument collapses if a finite-perimeter set has singular boundary points outside its reduced boundary: the assumption $H^n(\mathrm{Clos}(\partial^*E)\setminus\partial^*E)=0$ is what lets the proof replace $E$ by an open set with the same essential boundary and control the singular part.

Editorial extensions

If this is right

  • Volume-constrained critical points of any $\mathscr{C}^3$ elliptic anisotropic perimeter, among finite-perimeter sets with the boundary-closure condition, are finite unions of disjoint equal-radius Wulff shapes.
  • The Heintze-Karcher inequality is rigid: its equality cases are exactly Wulff-shape unions with radii at least $n/c$, so any set achieving equality is explicitly classified.
  • Smooth-boundary rigidity is recovered: a smooth closed hypersurface with constant positive anisotropic mean curvature must be a Wulff sphere.
  • The Lusin (N) property for anisotropic normal bundles is established for a broad class of sets, providing a tool for further measure-theoretic inequalities.
  • Any set satisfying the hypotheses whose volume exceeds the Heintze-Karcher bound cannot be a volume-constrained critical point, giving an explicit volume obstruction.

Reading between the lines

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

  • The condition $H^n(\mathrm{Clos}(\partial^*E)\setminus\partial^*E)=0$ is used to replace $E$ by an open set with the same essential boundary; it is plausible that a finer argument could remove it, which would settle the unrestricted finite-perimeter conjecture.
  • A quantitative stability statement should follow from the same proof: sets with small Heintze-Karcher deficit should be close in $L^1$ to a finite union of Wulff shapes, with a distance bound controlled by the deficit and the curvature bounds.
  • The level-set and normal-bundle machinery may extend to inequalities involving prescribed nonconstant anisotropic mean curvature or weighted perimeters, not only the constant-bound case treated here.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 5 minor

Summary. This paper establishes an anisotropic Heintze-Karcher inequality for finite-perimeter sets E in R^{n+1} whose anisotropic mean curvature is bounded between 0 and c and whose reduced boundary satisfies H^n(Clos(∂*E)\∂*E)=0: the inequality asserts L^{n+1}(E) ≤ n/(n+1) ∫_{∂E} 1/|hF| dH^n, with equality precisely when E is, up to a null set, a finite union of disjoint open Wulff shapes of radii at least n/c. The proof develops an anisotropic normal bundle, an anisotropic nearest-point projection, a Lusin (N) property for (n,h)-sets, and an anisotropic Steiner formula, and uses these tools to reduce the problem to the analysis of level sets of the anisotropic distance function. As a corollary, the paper claims that volume-constrained critical points of the anisotropic perimeter among finite-perimeter sets with the same boundary-closure hypothesis are finite unions of equal-radius Wulff shapes.

Significance. If correct, the main theorem substantially extends the isotropic result of Delgadino and Maggi to elliptic integrands of class C^3 and gives a positive answer to Maggi's conjecture in the restricted class of sets satisfying H^n(Clos(∂*E)\∂*E)=0. The proof is original and carefully structured; it leans explicitly on prior structural results ([33], [34], [31], [2], [11]) and introduces new tools (the anisotropic normal bundle and the anisotropic Steiner formula) that are likely to be of independent use. The equality analysis is detailed, and the paper is commendably explicit about the constants and hypotheses, with no free parameters fitted to the conclusion. The main caveat is that the proof of the central corollary, Corollary 6.8, contains a genuine gap (see major comment); additionally, the full conjecture without the boundary-closure hypothesis remains open, and the paper should state this limitation more prominently than it currently does.

major comments (1)
  1. [Section 6, proof of Corollary 6.8] The proof defines f_t(x) = (L(E)/L(h_t(E)))^(1/(n+1)) h_t(x) and claims that (27) applies to f_t because L(f_t(E)) = L(E) for every t. This is not justified: for a local variation h_t in the sense of Definition 6.6, h_t is the identity outside a compact set K, but f_t(x) = c_t x outside K with c_t = (L(E)/L(h_t(E)))^(1/(n+1)) generally different from 1. Hence {x : f_t(x) ≠ x} is unbounded and f_t is not a local variation, so the volume-constrained stationarity condition (27) does not apply. The corollary can be repaired by the standard Lagrange multiplier argument: (27) implies δP_F = λ δL for all compactly supported variations, and testing a sequence of radial cut-offs of the homothety vector field gives λ = n P_F(E)/((n+1)L(E)); then hF(V,x) = -λ n(E,x), so 0 < -hF·n = λ and equality holds in (17), allowing Theorem 6.4 to be invoked. This replacement should be incorporated.
minor comments (5)
  1. [Proof of Corollary 6.7] In the proof, 'a′(0)' should be 'p′(0)'.
  2. [Statement of Theorem 6.4] The text 'finite uni on' contains a typo and should read 'finite union'.
  3. [Title page] The author byline contains a garbled string 'S/suppress lawomir Kolasi´ nski'; it should read 'Sławomir Kolasiński'.
  4. [Notation throughout] The expression H^n(Clos(∂*E) ∼ ∂*E) uses '∼' for set difference; the notation should be clarified (e.g., using '\setminus') to avoid confusion with asymptotic equivalence.
  5. [Introduction] The introduction should state explicitly that Maggi's conjecture is proved only under the additional hypothesis H^n(Clos(∂*E)\∂*E)=0, which is not part of the original conjecture; the abstract does state this, but a remark in the introduction would make the scope clearer.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the anisotropic Heintze–Karcher inequality and Wulff rigidity are derived in-text from structural lemmas; self-citations are not load-bearing analogues of the target result.

full rationale

The paper's central result, Theorem 6.4, is an anisotropic Heintze–Karcher inequality with equality characterization. The proof does not take this inequality or the Wulff rigidity as an input: the main argument uses the anisotropic distance function, the Lusin (N) property proved in Section 4, the anisotropic Steiner formula in Section 5, and a Montiel–Ros area estimate in Section 6. The papers cited from the same authors ([31], [33], [34], [11]) supply prior structural lemmas—open representatives, distance-function/reach properties, and the weak maximum principle/(n,h)-set machinery—whose statements do not contain the target inequality or the Wulff uniqueness conclusion. The equality case is derived from equality in (17), not assumed. No parameter is fitted to the conclusion. The only notable issue is a proof gap in Corollary 6.8, where the rescaled map f_t is not a local variation in the sense of Definition 6.6 because it differs from the identity on an unbounded set; this is a correctness concern, not a circularity, and does not make the theorem an input to itself.

Assumptions & free parameters 0 free parameters · 6 assumptions · 2 invented entities

No empirical parameters are fitted; the theorem's constants c and n are hypotheses, and no free parameter is adjusted to data. The proof rests on standard geometric measure theory and on a set of external structural theorems, which are listed above. The invented entities are mathematical constructions introduced to make the proof work; they carry no independent empirical evidence because none is needed.

assumptions (6)
  • standard math Federer's geometric measure theory framework: area and coarea formulas, approximate tangent spaces, rectifiability, and varifold first variation
    Used throughout as background, with specific references to [17] and [3].
  • domain assumption Allard's regularity theorem for integral varifolds with controlled first variation with respect to an elliptic integrand
    Invoked in Claim 2 of Theorem 6.4 to obtain C^{1,α}, then C^{2,α}, regularity at H^n almost every boundary point.
  • domain assumption Weak maximum principle and area blow-up set characterization for (n,h)-sets from [11, Theorem 3.4]
    Used in Lemma 4.5 to identify spt of the varifold norm as an (n,h)-set, which is the starting point for the Lusin (N) condition.
  • domain assumption Santilli's open-representative lemma [33, Lemma 2.2]
    Replaces the finite-perimeter set E by an open set Ω with L^{n+1} difference zero and H^n(∂Ω ~ ∂*Ω)=0. This is where the boundary-closure hypothesis enters.
  • domain assumption Distance-function and normal-bundle calculus for arbitrary closed sets from [31] and [34], including approximate principal curvatures and bilipschitz level-set properties
    Provides the technical backbone of Sections 2, 4, and 5: regularity of level sets, approximate differentiability of projections, and the Lusin condition machinery.
  • domain assumption Federer's regularity theorem [17, 5.2.15], as modified in Remark 6.1 to upgrade C^{1,α} to C^{2,α} solutions of the anisotropic Euler-Lagrange equation
    The modification is sketched rather than fully proved, making this a load-bearing external regularity assertion.
invented entities (2)
  • Generalized anisotropic unit normal bundle N^F(A)
    purpose: Groups pairs (a, grad F(u)) of boundary point and anisotropic normal direction to support the Lusin condition, the Steiner formula, and the reach arguments.
    Mathematical definition introduced in Definition 5.1; it is justified internally by the proofs that use it, not by empirical evidence.
  • Anisotropic reach function reach^F(A) and anisotropic nearest-point projection ξ^F_A
    purpose: Control the size of the tubular neighborhood in the F*-metric where the projection is single-valued; used in Claims 5 and 6 of Theorem 6.4.
    Definitions 2.36 and 5.5 generalize the isotropic reach and are validated internally through the Steiner formula and level-set arguments.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Uniqueness of critical points of the anisotropic isoperimetric problem for finite perimeter sets." pith.science (2026). https://pith.science/paper/VSKS6GTB

@misc{pith2026190809795,
  author       = {Pith},
  title        = {Pith review of: Uniqueness of critical points of the anisotropic isoperimetric problem for finite perimeter sets},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VSKS6GTB}},
  note         = {Machine review of arXiv:1908.09795}
}
abstract

Given an elliptic integrand of class $ \mathscr{C}^{3} $, we prove that finite unions of disjoint open Wulff shapes with equal radii are the only volume-constrained critical points of the anisotropic surface energy among all sets with finite perimeter and reduced boundary almost equal to its closure.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

37 extracted references · 36 canonical work pages

  1. [33]

    Uniqueness of singular convex hypersurfaces with lower bounded k-th mean curvature

    Mario Santilli. The Heintze-Karcher inequality for sets of finite pe rimeter and bounded mean curvature. arXiv e-prints , page arXiv:1908.05952, Aug 2019

    work page Pith review arXiv 1908

  2. [34]

    The spherical image of singular varieties of bounded mean curvature

    Mario Santilli. The spherical image of singular varieties of bounded mean curvature. arXiv e-prints, page arXiv:1903.10379, Mar 2019

    work page Pith review arXiv 1903

  3. [31]

    Fine properties of the curvature of arbitrary closed sets

    Mario Santilli. Fine properties of the curvature of arbitrary clos ed sets. arXiv e-prints , page arXiv:1708.01549, August 2017

    work page Pith review arXiv 2017

  4. [2]

    William K. Allard. An integrality theorem and a regularity theorem for surfaces whose first variation with respect to a parametric elliptic integrand is controlled. Proceedings of Symposia in Pure Mathematics. Geometric Measure Theory and the Calcu lus of Variations , 44, 1986

    work page 1986

  5. [11]

    The Area Blow Up set for bounded mean curvature submanifolds with respect to elliptic surface energy functionals

    Guido De Philippis, Antonio De Rosa and Jonas Hirsch. The Area Blow Up set for bounded mean curvature submanifolds with respect to elliptic surface energ y functionals. Discrete Contin. Dyn. Syst. - A. , page arXiv:1901.03514, Jan 2019

    work page Pith review arXiv 1901

  6. [1]

    Alexandrov

    Aleksandr D. Alexandrov. Uniqueness theorems for surfaces in the large. V. Vestnik Leningrad. Univ., 13(19):5–8, 1958

    work page 1958

  7. [3]

    William K. Allard. On the first variation of a varifold. Ann. of Math. (2) , 95:417–491, 1972

    work page 1972

  8. [4]

    Frederick J.jun. Almgren. Almgren’s big regularity paper.Q-valued functions minimizing Dirich- let’s integral and the regularity of area-minimizing rectifiable currents up to codimension 2. Edited by V. Scheffer and Jean E. Taylor. Singapore: World Scientific, 2000

    work page 2000

Show all 37 references
  1. [5]

    Functions of bounded variation and free discon- tinuity problems

    Luigi Ambrosio, Nicola Fusco, and Diego Pallara. Functions of bounded variation and free discon- tinuity problems . Oxford Mathematical Monographs. The Clarendon Press, Oxfor d University Press, New York, 2000

    2000

  2. [6]

    Brothers and Frank Morgan

    John E. Brothers and Frank Morgan. The isoperimetric theorem for general integrands. Michigan Math. J. , 41(3):419–431, 1994

    1994

  3. [7]

    Bubbling with L2-almost constant mean curvature and an Alexandrov-type theor em for crystals Arch

    Matias Gonzalo Delgadino, Francesco Maggi, Cornelia Mihaila, and Ro bin Neumayer. Bubbling with L2-almost constant mean curvature and an Alexandrov-type theor em for crystals Arch. Rat. Mech. Anal. , 230(3): 1131–1177, 2018

    2018

  4. [8]

    Alexandrov’s theo rem revisited

    Matias Gonzalo Delgadino and Francesco Maggi. Alexandrov’s theo rem revisited. Version 1 of Arxiv: 1711.07690v1 , 2017

    2017 arXiv

  5. [9]

    Alexandrov’s theo rem revisited

    Matias Gonzalo Delgadino and Francesco Maggi. Alexandrov’s theo rem revisited. Anal. PDE , 12(6):1613–1642, 2019

    2019

  6. [10]

    Rectifi ability of varifolds with locally bounded first variation with respect to anisotropic surface e nergies

    Guido De Philippis, Antonio De Rosa and Francesco Ghiraldin. Rectifi ability of varifolds with locally bounded first variation with respect to anisotropic surface e nergies. Comm. Pure Appl. Math., 71(6):1123–1148, 2018

    2018

  7. [12]

    Regularity of free bounda ries in anisotropic capillarity problems and the validity of Youngs law

    Guido De Philippis and Francesco Maggi. Regularity of free bounda ries in anisotropic capillarity problems and the validity of Youngs law. Arch. Rat. Mech. Anal. , 216(2):473–568, 2015

    2015

  8. [13]

    Minimization of anisotropic energies in classes of rectifiable varifolds

    Antonio De Rosa. Minimization of anisotropic energies in classes of rectifiable varifolds. SIAM J. Math. Anal. , 50(1):162–181, 2018

    2018

  9. [14]

    Absence of bubbling phen omena for non convex anisotropic nearly umbilical and quasi Einstein hypersurfaces

    Antonio De Rosa and Stefano Gioffr` e. Absence of bubbling phen omena for non convex anisotropic nearly umbilical and quasi Einstein hypersurfaces. arXiv e-prints , page arXiv:1803.09118, Mar 2018

    2018 arXiv

  10. [15]

    Quantitative Stability for Anisotropic Nearly Umbilical Hypersurfaces

    Antonio De Rosa and Stefano Gioffr` e. Quantitative Stability for Anisotropic Nearly Umbilical Hypersurfaces. J. Geom. Anal. , 29(3):2318–2346, 2019

    2019

  11. [16]

    Curvature measures

    Herbert Federer. Curvature measures. Trans. Amer. Math. Soc. , 93:418–491, 1959

    1959

  12. [17]

    Geometric measure theory

    Herbert Federer. Geometric measure theory . Die Grundlehren der mathematischen Wis- senschaften, Band 153. Springer-Verlag New York Inc., New York , 1969. 25

    1969

  13. [18]

    A uniqueness proof for the Wulff theorem

    Irene Fonseca and Stefan M¨ uller. A uniqueness proof for the Wulff theorem. Proc. Roy. Soc. Edinburgh Sect. A , 119(1-2):125–136, 1991

    1991

  14. [19]

    Surface Evolution Equations : a level set metho d

    Yoshikazu Giga. Surface Evolution Equations : a level set metho d. Hokkaido University technical report series in mathematics , 71, 1, 2002

    2002

  15. [20]

    Uniqueness of constant weakly anis otropic mean curvature im- mersion of the sphere S2 in R3

    Yoshikazu Giga and Jian Zhai. Uniqueness of constant weakly anis otropic mean curvature im- mersion of the sphere S2 in R3. Adv. Differential Equations , 14(7-8):601–619, 2009

    2009

  16. [21]

    Compact embedd ed hypersurfaces with constant higher order anisotropic mean curvatures

    Yijun He, Haizhong Li, Hui Ma, and Jianquan Ge. Compact embedd ed hypersurfaces with constant higher order anisotropic mean curvatures. Indiana Univ. Math. J. , 58(2):853–868, 2009

    2009

  17. [22]

    Does polynom ial parallel volume imply con- vexity? Math

    Matthias Heveling, Daniel Hug, and G¨ unter Last. Does polynom ial parallel volume imply con- vexity? Math. Ann. , 328(3):469–479, 2004

    2004

  18. [23]

    A local Steiner-ty pe formula for general closed sets and applications

    Daniel Hug, G¨ unter Last, and Wolfgang Weil. A local Steiner-ty pe formula for general closed sets and applications. Math. Z. , 246(1-2):237–272, 2004

    2004

  19. [24]

    Linear algebra

    Serge Lang. Linear algebra. Undergraduate Texts in Mathematics. Springer-Verlag, New Yor k, third edition, 1987

    1987

  20. [25]

    Critical and almost-critical points in isoperime tric problems

    Francesco Maggi. Critical and almost-critical points in isoperime tric problems. Oberwolfach Rep., 35:34–37, 2018

    2018

  21. [26]

    Pointwise differentiability of higher order for sets

    Ulrich Menne. Pointwise differentiability of higher order for sets. Ann. Global Anal. Geom. , 55(3):591–621, 2019

    2019

  22. [27]

    Milman and Gideon Schechtman

    Vitali D. Milman and Gideon Schechtman. Asymptotic theory of finite-dimensional normed spaces, volume 1200 of Lecture Notes in Mathematics . Springer-Verlag, Berlin, 1986. With an appendix by M. Gromov

    1986

  23. [28]

    Compact hypersurfaces : the Alexandrov theorem for higher order mean curvatures

    Sebasti´ an Montiel and Antonio Ros. Compact hypersurfaces : the Alexandrov theorem for higher order mean curvatures. In Differential geometry , volume 52 of Pitman Monogr. Surveys Pure Appl. Math. , pages 279–296. Longman Sci. Tech., Harlow, 1991

    1991

  24. [29]

    Planar Wulff shape is unique equilibrium

    Frank Morgan. Planar Wulff shape is unique equilibrium. Proc. Amer. Math. Soc. , 133(3):809– 813, 2005

    2005

  25. [30]

    Tyrrell Rockafellar

    R. Tyrrell Rockafellar. Convex analysis . Princeton Mathematical Series, No. 28. Princeton University Press, Princeton, N.J., 1970

    1970

  26. [32]

    Rectifiability and approximate differentiability of highe r order for sets

    Mario Santilli. Rectifiability and approximate differentiability of highe r order for sets. Indiana Univ. Math. J. , 68:1013–1046, 2019

    2019

  27. [35]

    Quadratic tilt-excess decay and strong max imum principle for varifolds

    Reiner Sch¨ atzle. Quadratic tilt-excess decay and strong max imum principle for varifolds. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) , 3(1):171–231, 2004

    2004

  28. [36]

    Jean E. Taylor. Unique structure of solutions to a class of none lliptic variational problems. In Differential geometry (Proc. Sympos. Pure. Math., Vol. XXVI I, Stanford Univ., Stanford, Calif., 1973), Part 1 , pages 419–427, 1975

    1973

  29. [37]

    Zur Frage der Geschwindigkeit des Wachsturms un d der Au߬ osung der Kristall߬ achen.Z

    George Wulff. Zur Frage der Geschwindigkeit des Wachsturms un d der Aufl¨ osung der Kristallfl¨ achen.Z. Kristallogr. , 34:449–530, 1901. 26

    1901

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.