The one-dimensional equilibrium shape of a crystal

Emanuel Indrei

REVIEW 3 major objections 5 minor 1 cited by

The one-dimensional equilibrium shape of a crystal

T0 review · 3 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read In one dimension, convex sub-level sets of the potential force every mass-constrained free-energy minimizer to be an interval.

desk verdict A clean one-dimensional theorem with a false step in the existence proof; the result is probably right and the paper is worth engaging. read the letter →

arxiv 2501.07900 v1 pith:XLCYZSYI submitted 2025-01-14 math-ph math.APmath.CAmath.DGmath.MP

Emanuel Indrei This is my paper · ORCID

classification math-phmath.APmath.CAmath.DGmath.MP MSC 49Q2049J45

keywords freeenergyminimizationcrystalequilibriumshapeconvexsub-levelsetsone-dimensionalvariationalproblemrearrangementinequalityoptimaltransportintervalminimizercalculusofvariations

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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 answers a one-dimensional version of a classical variational question: if a crystal of fixed mass minimizes free energy under a potential with convex sub-level sets, must the minimizing shape be convex? The author's theorem says yes in one dimension: under $g(0)=0$, $g\ge 0$, and convex sub-level sets, the minimum is attained and every minimizer is an interval. The argument first shows that convex sub-level sets are equivalent to the potential being monotone on each half-line, then proves a rearrangement inequality that any candidate set has energy at least that of an interval of the same mass. This reduces the whole problem to choosing a translation, and the optimal translation $\alpha$ satisfies $g(\alpha+m)=g(\alpha)$.

What carries the argument

Two claims carry the argument. Claim 1: the sub-level sets $\{g<t\}$ are convex if and only if $g$ is non-decreasing on $[0,\infty)$ and non-increasing on $(-\infty,0]$. Claim 2: for any set $E$ with $|E|<\infty$, $\int_{E_+} g\,dx \ge \int_0^{|E_+|} g\,dx$ and $\int_{E_-} g\,dx \ge \int_{-|E_-|}^0 g\,dx$; therefore the energy of $E$ is at least the energy of the interval $(-|E_-|,|E_+|)$. The proof of Claim 2 is given twice: once by approximating $E_+$ with disjoint intervals and translating them leftward, and once by an optimal transport map that pushes the excess measure left while $g$ is monotone. The minimization then depends only on the one-variable function $a \mapsto 2+\int_a^{a+m} g(x)\,dx$, whose critical points satisfy $g(\alpha+m)=g(\alpha)$.

What would settle it

Set $g(0)=0$ and $g(x)=1$ for $x\neq 0$, and fix $m>0$. Then $E((0,m)+k)=2+m$ for every integer $k$, so the translations $a_k=k$ are minimizing with $|a_k|\to\infty$; this directly refutes the boundedness assertion used to prove existence. Any complete proof must add a coercivity assumption or replace that compactness step.

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Extended reading notes

Core claim

The paper's central claim is Theorem 1.1: for $n=1$, $m\in(0,\infty)$, $g(0)=0$, $g\ge 0$, and convex sub-level sets $\{g<t\}$, the infimum of $E(E)=\mathcal H^0(\partial E)+\int_E g(x)\,dx$ over sets of measure $m$ is attained, and every minimizer is convex. The author's argument concludes that a minimizer must be one of the four intervals $(0,m)+\alpha$, $[0,m)+\alpha$, $(0,m]+\alpha$, or $[0,m]+\alpha$, with $\alpha$ determined by the first-order condition $g(\alpha+m)=g(\alpha)$. The interval reduction comes from the rearrangement inequality in Claim 2: the potential energy of any set is no smaller than that of the interval $(-|E_-|,|E_+|)$, which has boundary cost $2$. The theorem is presented as an extension of prior results that required coercivity, radial symmetry, or higher regularity of the potential.

Load-bearing premise

In the existence part of the proof, Section 2 assumes that any minimizing sequence of interval translations has bounded positions; this fails for potentials that are zero at the origin and positive and constant elsewhere, because intervals far away achieve the same energy, so the compactness argument needs an added coercivity assumption or an alternative mechanism.

Editorial extensions

If this is right

Any minimizer is an interval of length $m$, so in one dimension the equilibrium crystal has exactly two boundary points.
The optimal translation is computable from the potential: it solves $g(\alpha+m)=g(\alpha)$; for even monotone potentials it is $\alpha=-m/2$.
The energy comparison reduces to one variable, making the constrained minimization a root-finding problem rather than a shape problem.
When $g$ is coercive ($g(x)\to\infty$ as $|x|\to\infty$), the compactness step needed for existence is valid, so the full theorem applies in that class.

Reading between the lines

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

For a potential that is zero at the origin and positive and flat elsewhere, such as $g(0)=0$ and $g(x)=1$ for $x\neq 0$, the interval energy is the same at arbitrarily large translations, so the boundedness step in the existence proof fails; the classification of minimizers as intervals still holds whenever a minimizer exists.
The optimal-transport proof of the rearrangement estimate suggests a quantitative stability statement: the energy gap between a set and its interval rearrangement should be controlled by how much $g$ grows under the leftward transport, giving control on the distance from near-minimizers to intervals.
In a thin strip with a potential that is large outside the strip, the one-dimensional theorem plausibly survives as a limit: as the strip width goes to zero, minimizers converge to the intervals described here.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Colleague—

The short version: this paper proves the natural 1D version of Almgren's problem—if g(0)=0, g≥0, and the sublevel sets {g

The soft spot is real and exactly where the stress-test puts it. In Section 2, after reducing to intervals I+a=(a,a+m), the proof asserts that any minimizing sequence (a_k) must be bounded because 'monotonicity yields sup_k |a_k| < ∞'. This is false. Take g(x)=1 for x≠0, g(0)=0. The sublevel sets are {0} and R, so the hypotheses hold, and for every a the integral ∫_a^{a+m} g is m, so E(I+a)=2+m is constant. The sequence a_k=k is minimizing and unbounded. The displayed contradiction in the paper relies on the strict inequality lim_l ∫_{I+a_{k_l}} g > ∫_0^m g, which fails here because both sides equal m. So the existence part of Theorem 1.1 is not established as written.

That said, the flaw is repairable and the theorem is very likely correct. If a minimizing sequence goes to +∞, monotonicity gives E(I+a) ≥ E(I) for every a>0, so a=0 already attains the infimum; the case a→−∞ is symmetric and gives the interval (−m,0). The convexity of minimizers follows from the rearrangement inequality once existence is granted. There is also a minor slip in the final paragraph where 'since H0(∂E)≥2, E is an interval' is not valid for arbitrary finite-perimeter sets, but Claim 2 (the mass rearrangement inequality) immediately supplies the correct argument, so this is not a serious issue.

Who is this for? People working in the calculus of variations, isoperimetric inequalities, or crystal shapes will find it a clean, quotable result in the one-dimensional base case. It deserves a serious referee: the theorem is new, the proof strategy is sound in its main line, and the existence gap is a few lines to fix. I'd send it to review, with a request that the author repair the boundedness argument and acknowledge the counterexample.

Referee Report

3 major / 5 minor

Summary. The paper studies the one-dimensional Almgren problem: minimize the free energy E(E) = H^0(∂E) + ∫_E g dx over sets E ⊂ R of prescribed measure m, where g ≥ 0, g(0)=0, and all sublevel sets {g < t} are convex. Theorem 1.1 claims that minimizers exist and are intervals. The proof first shows (Claim 1) that the sublevel-set condition is equivalent to g being nondecreasing on [0,∞) and nonincreasing on (−∞,0], then proves (Claim 2) a rearrangement inequality showing that for any set E, the interval with the same mass on the left and right of the origin has no larger potential energy. From this, the author reduces the problem to minimizing a ↦ E((0,m)+a), and then attempts to show that a minimizing sequence of translations is bounded. A separate optimal-transport proof of Claim 2 is also included, and Section 3 discusses identifying the optimal translation α.

Significance. If Theorem 1.1 is established, it settles the natural one-dimensional version of the Almgren convexity problem under very mild assumptions, and it gives a satisfying explanation of why convex minimizers should appear in one dimension. The paper's core rearrangement inequality (Claim 2) is correct, self-contained, and is the right key step; the optimal transport formulation of that inequality is elegant and potentially useful beyond this setting. The manuscript also correctly identifies that coercivity of g is not necessary for existence. However, the proof of existence contains a false boundedness claim, so the main theorem is not established as written. The convexity conclusion is likely salvageable with a modest repair, but the current version needs substantial correction before the result can be accepted.

major comments (3)

[Section 2, paragraph beginning 'If a_k are numbers such that ...'] The assertion 'monotonicity yields sup_k |a_k| < ∞' is false. Let g(x)=1 for x≠0 and g(0)=0. Then the sublevel sets {g<t} are {0} for 0<t≤1 and R for t>1, so the hypotheses of Theorem 1.1 hold. For every a, ∫_a^{a+m} g(x) dx = m, hence E((0,m)+a)=2+m for all a. Therefore a_k=k is a minimizing sequence for inf_a E((0,m)+a) with |a_k| → ∞, contradicting the claimed uniform boundedness. The proof's contradiction uses the strict inequality lim_l ∫_{I+a_{k_l}} g > ∫_0^m g; in this example the two integrals are equal for every a, so the argument collapses. Since this boundedness is used to extract a convergent subsequence and prove existence, the existence half of Theorem 1.1 is not established as written. The claim can likely be repaired by treating subsequences tending to +∞ or −∞ separately, but the repair is not present.
[Section 2, final paragraph of the proof of Theorem 1.1] The inference 'if E ⊂ {g=0}, then since H^0(∂E) ≥ 2, E is an interval' is not valid for arbitrary finite-perimeter sets. For instance, the union of two disjoint intervals of total length m has H^0(∂E)=4 and is not an interval. The conclusion that any minimizer is an interval can nevertheless be obtained directly from Claim 2, which shows that any disconnected E is strictly dominated by an interval of the same mass (once the existence of a minimizer is known). The proof should be restated to use Claim 2 instead of the invalid inference from the boundary count.
[Section 3, first-order condition and examples] The derivation g(α+m)=g(α) from differentiating G(a+m)−G(a) presupposes differentiability (or at least absolute continuity) of G, which is not assumed in Theorem 1.1. Moreover, the examples impose strict monotonicity and continuity hypotheses that are not part of the theorem. The paragraph asserting 'one always finds a minimizer E_m with 0 ∈ E_m' and that 'E_m is a minimizer as well' is not proved in the stated generality and appears to conflict with cases where g vanishes on a half-line. These statements should either be proved under the actual assumptions or explicitly presented as heuristic.

minor comments (5)

[Throughout] There are numerous typographical errors, including 'countin g' in the introduction, 'th e', 'suppo sing', and 'theo rem'. The text would benefit from a careful proofreading pass.
[Section 2, first paragraph] The notation 'E0' is undefined; the text should read 'if E^0 = ∅' for the interior, and the subsequent inclusion should be written as 'E ⊂ ∂E ∪ E^0'.
[Section 2, Claim 2 proof] The dominated convergence argument requires that g be locally integrable (or at least that the relevant integrals are finite), but the theorem's hypotheses only state g≥0 with convex sublevel sets. The assumptions should explicitly include measurability and local integrability of g.
[Section 2.2] Claim 2 is stated and proved a second time with the same numbering as in Section 2; this duplicate numbering is confusing and should be renumbered.
[Section 3] The notation 'H^0(∂(−|E_−|,|E_+|)) = 2' should be clarified with parentheses, e.g., 'H^0(∂((−|E_−|,|E_+|))) = 2'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Theorem 1.1 is derived from the stated hypotheses without load-bearing self-citation or fitted inputs.

full rationale

The proof of Theorem 1.1 is self-contained. Claim 1 derives monotonicity of g on each half-line from convexity of the sublevel sets {g<t}; Claim 2 uses that monotonicity to compare the potential energy of any admissible set with that of an interval of the same mass. The key reduction E(E) >= inf_a E(I+a) depends only on the elementary fact H0(dE) >= 2 for sets of positive finite measure, and the later identification of a minimizing translation is attempted through direct compactness and monotonicity arguments. The author's prior works are cited only in the introduction and remarks for context, comparison, and related results; none is invoked as a substitute for a step in the proof. There are no fitted parameters, no quantity is defined in terms of the conclusion, and no known result is merely renamed. The skeptical concern about the boundedness of minimizing translations is a potential mathematical gap in the existence argument, but it is a correctness issue, not a circularity: the claim is not equivalent to the theorem by construction, nor is it imported from a self-citation. Therefore the appropriate circularity score is 0.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

No free parameters or invented entities. The proof relies only on standard measure theory and the theorem's assumptions.

assumptions (3)

domain assumption g is real-valued with g(0)=0 and g>=0; convex sub-level sets imply g is monotone on each half-axis (Claim 1), hence locally integrable.
The theorem's assumptions. The proof uses integrals of g over bounded intervals and translates.
standard math Sets of finite perimeter in R have H^0(∂E) equal to the number of boundary points; candidates with finite energy have H^0(∂E)<∞.
The proof restricts to non-empty interior sets and uses H^0(∂E)>=2; the equality H^0(∂E)=2 for an interval is used in the comparison.
standard math Dominated and monotone convergence theorems apply to the approximations E^k_+ and truncations I_R.
Used in Claim 2 to pass from finite unions of intervals to arbitrary measurable E_+.

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Cite this review

Pith. "Pith review of The one-dimensional equilibrium shape of a crystal." pith.science (2026). https://pith.science/paper/XLCYZSYI

@misc{pith2026250107900,
  author       = {Pith},
  title        = {Pith review of: The one-dimensional equilibrium shape of a crystal},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XLCYZSYI}},
  note         = {Machine review of arXiv:2501.07900}
}

abstract

Optimizing the free energy under a mass constraint may generate a convex crystal subject to assumptions on the potential $g(0)=0$, $g \ge 0$. The general problem classically attributed to Almgren is to infer if this is the case assuming the sub-level sets of g are convex. The theorem proven in the paper is that in one dimension the answer is positive.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Free energy minimizers with radial densities: classification and quantitative stability
math.AP 2024-12 conditional novelty 7.0 of 10

Centered spheres minimize the weighted free energy for all volumes only under extra monotonicity of both weights; otherwise the second-variation condition psi''+g' >= 0 can fail to select global minimizers.

Reference graph

Works this paper leans on

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [1]

Shrey Aryan and Lauro Silini, Free energy minimizers with radial densities: classification and quantitative stability, arXiv:2412.03997 (2024)

work page Pith review arXiv 2024

[2] [2]

Konstantinos Bessas, Matteo Novaga, and Fumihiko Onoue, On the shape of small liquid drops minimizing nonlocal energies, ESAIM Control Optim. Calc. Var. 29 (2023), Paper No. 86, 26. 4674821

work page 2023

[3] [3]

P. Curie, Sur la formation des cristaux et sur les constantes capillaires de leurs different faces, Bulletin de la Societe Francaise de Mineralogie et de Cristallographie 8 (1885), 145--150

work page

[4] [4]

Guido De Philippis and Michael Goldman, A two-point function approach to connectedness of drops in convex potentials, Comm. Anal. Geom. 30 (2022), no. 4, 815--841. 4545852

work page 2022

[5] [5]

Irene Fonseca and Stefan M\" u ller, A uniqueness proof for the W ulff theorem , Proc. Roy. Soc. Edinburgh Sect. A 119 (1991), no. 1-2, 125--136. 1130601

work page 1991

[6] [6]

Figalli and F

A. Figalli and F. Maggi, On the shape of liquid drops and crystals in the small mass regime, Arch. Ration. Mech. Anal. 201 (2011), no. 1, 143--207. 2807136

work page 2011

[7] [7]

Irene Fonseca, The W ulff theorem revisited , Proc. Roy. Soc. London Ser. A 432 (1991), no. 1884, 125--145. 1116536

work page 1991

[8] [8]

2, 422--446

Alessio Figalli and Yi Ru-Ya Zhang, Strong stability for the W ulff inequality with a crystalline norm , Communications on Pure and Applied Mathematics 75 (2022), no. 2, 422--446

work page 2022