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The p-widths of a polygon

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Pith's one-line read Every p-width of a convex polygon is achieved by billiard trajectories.

desk verdict The reflection argument is clever and the explicit widths are nice, but the proof of the main billiard theorem is off by the missing h0 normalization factor. read the letter →

arxiv 2505.03047 v1 pith:PQ7BSFTL submitted 2025-05-05 math.DG

Otis Chodosh , Sithipont Cholsaipant This is my paper · ORCID
classification math.DG MSC 53C4258E1249Q2035J60
keywords p-widthsbilliardtrajectoriesconvexpolygonsAllen-Cahnequationmin-maxtheoryphasetransitionequilateraltrianglesquare
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

This paper proves that the p-widths of a convex polygon—a sequence of geometric invariants analogous to the Neumann Laplacian spectrum—are always realized by finite unions of billiard trajectories. For each p, the minimax length can be written exactly as the sum of lengths of finitely many straight segments that reflect off the polygon boundary. The proof routes the problem through Allen-Cahn phase transitions, reflecting critical points across the boundary to reduce the boundary case to an interior theorem. As concrete evidence, the paper computes the first four widths of the equilateral triangle inscribed in the unit circle and the first three widths of the square, matching explicit billiard configurations.

What carries the argument

The main mechanism is Allen-Cahn phase-transition min-max: p-widths are recovered as $\varepsilon\to 0$ limits of critical points of the double-well energy $E_\varepsilon[u]=\int(\frac{\varepsilon}{2}|\nabla u|^2+\frac{1}{\varepsilon}W(u))$, with Neumann boundary conditions. The load-bearing step is an even reflection across a boundary edge: reflecting a Neumann critical point produces a solution on the doubled domain whose index is at most $2p$, so the closed-surface billiard theorem applies and the limiting energy measure is a union of straight line segments. A monotonicity estimate at vertices, valid without boundary correction because polygon edges are straight and the gradient vanishes at convex corners, controls the mass near vertices. For the equilateral triangle, repeated reflections generate the full tessellation and force the billiards to unfold to straight lines; this yields the length set $\{\frac{3}{2}\sqrt{a^2+ab+b^2}:a,b\in\mathbb{Z}\}$ used in the explicit computations.

What would settle it

Run a numerical phase-field min-max simulation of the Allen-Cahn energy on a generic convex pentagon and inspect the limiting interface: if for some small $\varepsilon$ the minimizing sweepout converges to a configuration with an interior triple junction or a corner-touching segment that does not reflect specularly, the billiard decomposition would be false.

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Extended reading notes

Core claim

For $P$ a compact convex polygon in $\mathbb{R}^2$ and any $p=1,2,\ldots$, there exist finitely many billiard trajectories $\gamma_{p,1},\ldots,\gamma_{p,N(p)}$ in $P$, repetitions allowed, with total length equal to $\omega_p(P)$. The trajectories are allowed to terminate either orthogonally at the boundary or at a vertex, and otherwise follow specular reflection. The first width of any convex polygon equals its geometric width $W(P)$, the width of the narrowest slab containing it. For the equilateral triangle inscribed in the unit circle the paper obtains $\omega_1=\omega_2=3/2$, $\omega_3=3\sqrt{3}/2$, and $\omega_4=3$; for the square of side $\sqrt{2}$ it obtains $\omega_1=\sqrt{2}$, $\omega_2=2$, and $\omega_3=2\sqrt{2}$. In the triangle these values are realized respectively by a perpendicular bisector, the medial triangle, and two perpendicular bisectors; in the square by horizontal or vertical segments and a diagonal.

Load-bearing premise

The proof depends on the even reflection of Allen-Cahn critical points having index at most $2p$ and on a vertex monotonicity estimate with no boundary correction; if either fails, the limiting interfaces need not be line segments.

Editorial extensions

If this is right

  • For every convex polygon and every p, the variational minimax length can be certified by an explicit finite collection of reflecting straight segments.
  • The first p-width of a convex polygon is always the width of its narrowest enclosing slab, so one can read $\omega_1$ off from the polygon's geometry alone.
  • In an equilateral triangle, a single side or two sides cannot furnish a p-width; only full boundary-type billiards or configurations allowed by unfolding can.
  • The method extends, as the paper notes, to Riemannian surfaces whose boundary is piecewise totally geodesic with convex corners, replacing line segments by geodesic segments and allowing closed geodesics.
  • The computed values give nontrivial exact p-widths for polygons, providing benchmarks for the Weyl asymptotic and Lusternik-Schnirelman inequalities.

Reading between the lines

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

  • One can expect an algorithm: for a rational convex polygon, the finite graph of possible billiard trajectories with bounded length should make each p-width computable by solving a shortest-path or integer-optimization problem over reflection-unfolded straight lines; the paper's length-set lemma for the equilateral triangle is a prototype.
  • If the reflection-index bound survives perturbations, the same proof should give billiard decompositions for p-widths in higher-dimensional convex polytopes with totally geodesic faces, where the limiting objects would be unions of reflecting geodesic segments rather than planar billiards.
  • The vertex condition that billiards stop at corners suggests a spectral analogue: p-widths may encode the 'billiard graph' of a polygon, with corner hits acting as Dirichlet-like breaks; comparing $\omega_p$ for polygons that differ only by rounding corners could test how much of the p-width is carried by vertex-terminating trajectories.
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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

4 major / 5 minor

Summary. The paper studies Gromov–Guth p-widths of compact convex polygons in R^2. The main theorem (Theorem 1.3) asserts that every p-width of such a polygon is the sum of lengths of finitely many billiard trajectories. The proof follows the Allen–Cahn route of Chodosh–Mantoulidis [CM23], combined with an even reflection argument across boundary edges to reduce boundary bounces to interior line segments on a doubled domain. The paper also proves that the first width of any convex polygon equals its geometric width (Theorem 1.6), and computes the low widths of the equilateral triangle (p=1,...,4) and the square (p=1,...,3), with explicit billiard realizations. The exposition is generally clear, but the central argument contains a missing normalization constant h0 that affects the claimed equality in Theorem 1.3, and several analytic steps at polygonal corners and under reflection are either asserted without proof or only sketched.

Significance. If the main theorem is correct, it is a substantial step: it gives the first general billiard-realization result for p-widths in domains with non-smooth boundary, complementing the closed-surface theorem of [CM23] and the smooth-domain conjecture. The explicit computations for low p-widths of the equilateral triangle and square are concrete and genuinely useful, and the geometric-width result for the first width is clean and self-contained. The paper is also valuable for pointing out exactly why a boundary version of Allen–Cahn curvature estimates is not needed, via reflection. However, the central proof currently has a quantitative gap involving the heteroclinic energy h0, so the main theorem is not established as stated.

major comments (4)
  1. [Section 3, after Proposition 3.4] The proof of Theorem 1.3 misses the h0 normalization. Proposition 2.6 states that lim_{\epsilon\to0} \omega_{p,\epsilon}(P) = h0 \omega_p(P), where h0 is the heteroclinic energy. In Section 3 the potential is fixed to be W(t)=(1+\cos \pi t)/\pi^2; for this potential h0 = \int_{-1}^{1} \sqrt{2W(s)}\,ds = 8/\pi^2 \neq 1. The measure \mu is defined as the weak limit of the unnormalized energy densities, so its total mass is \lim \omega_{p,\epsilon}(P) = h0 \omega_p(P). Proposition 3.4 concludes \mu = \sum_j H^1\lfloor \eta_j, and the final grouping then gives \sum_j \operatorname{length}(\eta_j) = h0 \omega_p(P), not \omega_p(P). Thus Theorem 1.3 does not follow as stated. The discrepancy is quantitative: for the equilateral triangle, Theorem 1.8 claims \omega_1(T)=3/2, while the argument as written would produce a limiting measure of mass h0\cdot 3/2 = 12/\pi^2 \approx 1.216. This must be repaired, for example by normalizing the Allen–Cahn widths by h0^{-1} from the outset or by choosing a potential with h0=1.
  2. [Section 3, Lemma 3.3] The monotonicity formula at polygonal vertices is load-bearing but is asserted rather than proved. The lemma claims \mu_\epsilon(B_r(p)) \le Cr for p \in V with no boundary correction, citing [HT00] and [LPS24]. However, [LPS24] concerns smooth boundaries, and at a corner the Neumann condition does not give a smooth reflected problem; the statement 'no boundary correction is necessary' needs a detailed proof or a precise reference covering polygonal corners. This step is essential because it is used to conclude \mu(V)=0 and hence to discard vertex contributions when grouping segments into billiard trajectories.
  3. [Section 3, reflected index bound] The proof that the even reflection \tilde u_{\epsilon_j} has index at most 2p on B is too compressed to be verifiable as written. The argument invokes 'standard results' and compares Dirichlet and Neumann eigenvalue counts, but it does not spell out how the spectrum of the reflected operator on B is identified with the Dirichlet and Neumann spectra on B^+, nor how the L^2-orthogonality argument controls the full quadratic form. Since this index bound is what permits the application of [CM23, Theorem 1.2] to the reflected problem, a complete proof or a precise reference is needed.
  4. [Section 2.3, Proposition 2.6] Proposition 2.6 states that the comparison h0^{-1} \lim_{\epsilon\to0} \omega_{p,\epsilon}(K) = \omega_p(K) remains valid for polygonal regions, with only the remark that 'a careful examination of the proof shows' this. This is not a trivial extension, because the Allen–Cahn min-max theory near corners and the boundary condition require additional arguments; the rest of the paper depends on this comparison. The authors should either provide the extension in detail or cite a source that proves the comparison for Lipschitz polygonal domains with Neumann boundary conditions.
minor comments (5)
  1. [Section 1.1, Conjecture 1] In the displayed formula for the conjecture, the summand is written as \operatorname{length}(\gamma_{p,N(p)}) but should be \operatorname{length}(\gamma_{p,j}).
  2. [Section 2.3, definition of h0] The displayed definition of h0 appears to have a typo: it reads H'(t)^2 + W'(H(t)); the standard heteroclinic energy is \int (\tfrac12|H'|^2 + W(H))\,dt, and as written W'(H) integrates to zero by the traveling-wave equation. This should be corrected for clarity.
  3. [Section 4, T-billiard definition] The definition of a T-billiard trajectory switches notation between \ell_k and \eta_k in conditions (1") and (2"), which makes the definition harder to follow. Please use one symbol consistently.
  4. [Section 4, Proposition 4.2] The statement of Proposition 4.2 writes \omega_p(P) but the theorem is about the equilateral triangle T; the variable P here is either a typo or needs a line defining P=T.
  5. [Section 7.1] There is a minor typo: 'a the 2-sweepout' should be 'a 2-sweepout'.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: p-widths are min-max invariants and billiard lengths come from the Allen-Cahn limit interface via independent results. A separate h0 normalization gap blocks the stated equality in Theorem 1.3.

full rationale

The derivation is not circular at the level of definitions or fitting. p-widths are defined by Gromov-Guth min-max over sweepouts (Definition 2.2), and no parameter is fitted: the billiard lengths in Theorem 1.3 are obtained from the weak limit mu of the Allen-Cahn energy densities (Section 3), not imposed from the target value omega_p(P). Reliance on [CM23] is a self-citation, but it supplies an independent published classification result (p-widths of closed surfaces, and its local proof) whose stated assumptions do not include the polygon conclusion; per the rubric, such a citation is real evidence and does not raise the circularity score. The reflection argument and the index bound are local and do not presuppose the billiard decomposition. However, a normalization issue remains outside circularity: Proposition 2.6 fixes h0^{-1} lim_{epsilon to 0} omega_{p,epsilon}(P) = omega_p(P), and for W(t)=(1+cos(pi t))/pi^2 one has h0 = integral H'^2 = 8/pi^2. The measure mu in Section 3 is the unnormalized limit with mass lim omega_{p,epsilon} = h0 omega_p(P), so Proposition 3.4 gives sum length(eta_j) = h0 omega_p(P), not omega_p(P). The final sentence 'Grouping the eta_j into billiard trajectories ... this proves Theorem 1.3' omits this factor; this is a correctness/missing-factor gap, not a circular reduction.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claim rests on standard min-max and Allen-Cahn background plus several domain-specific assumptions. The most fragile are the monotonicity formula at polygonal corners and the extension of the Allen-Cahn width convergence to polygonal regions. No free parameters are fitted to data, and no new entities are postulated.

assumptions (5)
  • domain assumption Allen-Cahn min-max theory and the convergence h0^{-1} lim_{ε→0} ω_{p,ε}(K) = ω_p(K) for Lipschitz and polygonal domains
    Proposition 2.6 cites [Dey22] and [GG18] and asserts the extension to polygonal regions by 'careful examination of the proof'; this extension is not demonstrated in the paper.
  • domain assumption [CM23, Theorem 1.2]: the p-widths of a closed surface are realized by finite sums of lengths of closed geodesics
    Used as a black box in the reflection argument in Section 3 to conclude that the limiting energy measure on the doubled domain is a union of straight line segments.
  • domain assumption [DM24]: for a strictly convex smooth planar domain, the first width is achieved by a free boundary geodesic (a segment meeting the boundary orthogonally)
    Used in the proof of Theorem 1.6 to obtain the lower bound ω1(P) ≥ W(P).
  • ad hoc to paper Monotonicity formula for Allen-Cahn critical points at polygonal vertices with Neumann boundary holds without boundary correction
    Lemma 3.3 asserts r^{-1} μ_ε(B_r(p)) ≤ C for vertices p, citing [HT00] and [LPS24], but those results treat smooth boundaries; the corner case is not explicitly proven.
  • domain assumption Billiards of finite length cannot bounce infinitely many times into a convex corner
    Cited to [Lan23, Lemma 2.3] and used to group the limiting line segments into finite billiard trajectories.

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Pith. "Pith review of The p-widths of a polygon." pith.science (2026). https://pith.science/paper/PQ7BSFTL

@misc{pith2026250503047,
  author       = {Pith},
  title        = {Pith review of: The p-widths of a polygon},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PQ7BSFTL}},
  note         = {Machine review of arXiv:2505.03047}
}
abstract

The $p$-widths are a nonlinear analogue of the spectrum of the Laplacian. We prove that each $p$-width of a polygon in $\mathbb{R}^2$ is achieved by a union of billiard trajectories. We also compute the $p$-widths of the equilateral triangle for $p=1,\dots,4$ and square for $p=1,\dots,3$.

Figures

Figures reproduced from arXiv: 2505.03047 by the authors.

Figure 1
Figure 1. The definition of the map ϕ : ∂T × ∂T → Z1(T, ∂T) used to construct an optimal 2-sweepout of the equilateral triangle T [PITH_FULL_IMAGE:figures/full_fig_p013_1.png] view at source ↗
Figure 2
Figure 2. Deriving an upper bound for the length of ϕ when l1 ∩ l2 = p ∈ T. as p2 is similar). See [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 3
Figure 3. Deriving an upper bound for the length of ϕ when l1 ∩ l2 = p lies outside of T but on the same side of T as p1. Finally, we assume that l1 ∩ l2 = p ̸∈ T is on the opposite side of the edge Γ of T not containing p1, p2. See [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Deriving an upper bound for the length of ϕ when l1 ∩ l2 = p lies outside of T on the opposite side of T as p1, p2. This completes the proof. □ Lemma 6.2. ϕ is continuous. Proof. It suffices to check continuity at (p1, p2) where one of the pi (without loss of generalit…

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