Corner structure and chamberwise product topology for variable-length planar polygons

Definitions and mathematical statement

Holmes-Cerfon asks for the topology and volume of planar polygon configuration spaces when the labelled edge lengths satisfy $L_i-\epsilon\leq\ell_i\leq L_i+\epsilon$. To make one standard interpretation precise, take $n\geq3$, $\epsilon>0$, and $L_i>\epsilon$ for every $i$. Let \[ B=\prod_{i=1}^n[L_i-\epsilon,L_i+\epsilon],\qquad M_{L,\epsilon}=\left\{(z_1,\ldots,z_n)\in{\mathbb{C}}^n: \sum_i z_i=0,\ (|z_1|,\ldots,|z_n|)\in B\right\}/S^1, \] where $S^1$ acts by simultaneous rotation. Edge vectors describe labelled closed polygons modulo translations and orientation-preserving Euclidean motions; self-intersections are permitted, and reflections are not identified. The length map is $\pi([z])=(|z_i|)_i$, with fixed-length fiber $M_\ell=\pi^{-1}(\ell)$. These quotient and embedding conventions are not specified in the original question.

The source describes $M_{L,\epsilon}$ as a manifold with boundary and asks to understand its topology and volume. Simultaneous closed interval constraints naturally give corners; smoothness is not automatic without regularity assumptions. The supplied aid reports that vertex-genericity of $B$, meaning \[ \sum_{i=1}^n\sigma_i\ell_i\ne0 \quad\text{for every vertex }\ell\text{ of }B \text{ and every }\sigma\in\{-1,1\}^n, \] suffices for the nonempty configuration space to be a compact smooth manifold with corners of dimension $2n-3$. At collinear configurations on a positive-dimensional length face, a free radial edge variation supplies the derivative missing from angular variations; at vertices, the stated condition excludes collinear configurations.

For a generic central vector $L$, put \[ \delta(L)=\min_{\sigma\in\{-1,1\}^n}\left|\sum_i\sigma_iL_i\right|>0. \] The aid reports that $n\epsilon<\delta(L)$ gives a noncanonical diffeomorphism over the entire length box \[ M_{L,\epsilon}\cong M_L\times B, \] intertwining $\pi$ with projection to $B$. This is a chamberwise statement, not a description when the box crosses a signed-length wall.

For $n=3$, assume every length vector in $B$ satisfies the strict triangle inequalities. Then each fiber has two points, distinguished by orientation, and the aid reports two copies of $B$. A precise length-counting volume in this zero-dimensional-fiber case is \[ \int_B\#M_\ell\,{\,\mathrm{d}}\ell_1{\,\mathrm{d}}\ell_2{\,\mathrm{d}}\ell_3 =2(2\epsilon)^3=16\epsilon^3. \] If reflections are identified, there is one copy of $B$, and this volume is $8\epsilon^3$. These formulas concern length-counting measure, not an unspecified Riemannian quotient volume. For general $n$, fibers are positive dimensional and the source must specify a measure or metric before "volume" has a numerical meaning.

Short English statement

Study labelled planar polygons whose edge lengths vary independently in intervals. Under explicit genericity assumptions the supplied aid describes a compact manifold with corners and, within one length chamber, a product with the fixed-length polygon space. Volume depends on the chosen quotient and measure; the reported triangle formula counts orientation fibers over length space.

Sources

• Holmes-Cerfon, problem 4 in the AIM workshop Configuration spaces of linkages: https://aimath.org/pastworkshops/linkagesproblems.pdf. Canonical location: aim-geometry-notes.json, notes[190].
• Local registry data/unsolvedmath/problems.json, permanent ID 20001853, code catalogue entry; updated 14 August 2026. Statement status is reconstructed unverified. The recorded verification note reports inspection of PDF page 1 and reconstruction of $\epsilon$ and $\ell_i$ from extraction artifacts; no new PDF inspection is asserted here. The original literature field is empty.
• Supplied attempt 1: partial result, problem status at run open, rigor label proved, novelty confidence medium; current registry status partially solved. This research aid is machine generated, not peer reviewed, and requires independent verification before citation.