Cone Length: Analysis & Applications

• Cone length is a context-dependent invariant that quantifies complexity in diverse settings including geometric, topological, combinatorial, and algebraic frameworks.
• In flat cone geometry and hyperbolic surfaces, it measures geodesic lengths and boundary parameters, underpinning rigidity proofs and counting results with explicit bounds.
• In rational homotopy, Coxeter theory, and SOS decompositions, cone length serves as a structural metric to count cofibration stages, minimal representatives, or Gram ranks.

Searching arXiv for the cited papers to ground the article in current arXiv records. {"query":"id:(Fu, 2024) OR id:(Pan, 2017) OR id:(Parent et al., 14 Oct 2025) OR id:(Parkinson et al., 2021) OR id:(Allen et al., 2013) OR id:(Laplagne et al., 2020) OR id:(Janaswamy, 2023) OR id:(Gazwani et al., 14 Jun 2026) OR id:(Rukhovich, 2022) OR id:(Gheorghita et al., 2014)","max_results":10} Retrieving a few of the most directly relevant entries individually. {"query":"(Fu, 2024)","max_results":5} “Cone length” is a context-dependent term rather than a single invariant. In current research usage it can denote the Euclidean length of a trajectory in a flat cone metric, a generalized boundary assignment attached to a cone point on a hyperbolic cone surface, the minimal number of cofibration stages needed to build a space up to homotopy, the length of the unique minimal representative of a Coxeter cone type, or the minimal number of squares in a sum-of-squares decomposition (Fu, 2024, Pan, 2017, Parent et al., 14 Oct 2025, Parkinson et al., 2021, Laplagne et al., 2020). The common feature is structural rather than semantic uniformity: each usage measures complexity relative to a cone-based geometry, combinatorics, or homotopical construction.

1. Terminological range

The literature represented here uses the phrase in several non-equivalent ways.

Domain	Meaning of “cone length”	Representative source
Flat cone geometry	Euclidean length $|\gamma|$ of a trajectory or regular closed geodesic in the flat cone metric	(Fu, 2024)
Hyperbolic cone surfaces	Generalized boundary assignment $\lambda(\Delta)=-\theta$ for a cone point of angle $\theta$	(Pan, 2017)
Rational homotopy	Minimum number of cofibrations needed to build a space from a suspension by attaching suspensions	(Parent et al., 14 Oct 2025)
Coxeter groups	$L(T)=\ell(m_T)$, the length of the unique minimal representative of a cone type $T$	(Parkinson et al., 2021)
SOS cones	Minimal number of squares in an SOS decomposition, equal to minimal Gram rank	(Laplagne et al., 2020)

A common misconception is that “cone length” always refers to a metric length. That is correct in flat cone geometry and in several cone-associated PDE or EM settings, but it is incorrect in rational homotopy, Coxeter theory, and SOS geometry. Another source of ambiguity is terminology involving both “cone” and “length” without defining an invariant called cone length at all; for example, “the cone of Betti tables of finite length modules” concerns a rational polyhedral cone indexed by finite length modules rather than a quantity named cone length (Gheorghita et al., 2014).

2. Flat cone surfaces, self-intersection, and billiards

In the flat-geometric setting, a flat cone surface $X$ is a Riemann surface endowed with a flat metric away from finitely many conical singularities $x_1,\dots,x_n$, and near each $x_i$ the metric is locally isometric to a Euclidean cone of angle $2\pi(1-k_i)$, where $k_i$ is the curvature at $\lambda(\Delta)=-\theta$0. When the underlying surface is the sphere and all singularities are conical, $\lambda(\Delta)=-\theta$1 is a flat cone sphere; it is convex if all curvatures are positive, and the Gauss–Bonnet identity imposes $\lambda(\Delta)=-\theta$2 (Fu, 2024).

The basic length quantity is the flat-metric length $\lambda(\Delta)=-\theta$3 of a trajectory $\lambda(\Delta)=-\theta$4, where a trajectory is a finite-length geodesic segment whose interior avoids singularities. For a regular closed geodesic, the paper establishes lower bounds in terms of the self-intersection number

$\lambda(\Delta)=-\theta$5

On any unit-area flat cone surface, one has

$\lambda(\Delta)=-\theta$6

and for any saddle connection or regular closed geodesic,

$\lambda(\Delta)=-\theta$7

Here

$\lambda(\Delta)=-\theta$8

The constants depend on the flat metric through the relative systole, Delaunay circumscribed radii, and minimum angular deficit (Fu, 2024).

For convex flat cone spheres, the geometry becomes uniform after imposing two restrictions: a lower bound on the curvature gap

$\lambda(\Delta)=-\theta$9

and an upper bound on the number of singularities $\theta$0. On any unit-area convex flat cone sphere with $\theta$1,

$\theta$2

and for regular closed geodesics,

$\theta$3

The explicit constants are

$\theta$4

A further universal statement holds for every regular closed geodesic: $\theta$5 Combining this with the uniform upper bound of (Fu et al., 2023) yields two-sided comparability

$\theta$6

for regular closed geodesics on the same class of surfaces (Fu, 2024).

The proof mechanism is triangulation-based. A trajectory decomposes into threads across a geometric triangulation; a corner-switch lemma forces length accumulation in sufficiently long thread blocks, the width $\theta$7 of a Delaunay triangulation is controlled by $\theta$8 and circumscribed radii, and the combinatorial complexity satisfies

$\theta$9

Uniformity over moduli space is then achieved by a thick–thin decomposition and generalized Thurston surgeries on convex hulls of short forests (Fu, 2024).

These estimates are sharp enough to support counting theory. For unit-area convex flat cone spheres with $L(T)=\ell(m_T)$0 singularities and curvature gap $L(T)=\ell(m_T)$1, if $L(T)=\ell(m_T)$2 counts saddle connections of length at most $L(T)=\ell(m_T)$3 and $L(T)=\ell(m_T)$4 counts maximal families of parallel regular closed geodesics of length at most $L(T)=\ell(m_T)$5, then

$L(T)=\ell(m_T)$6

Via the doubling construction for a convex polygon $L(T)=\ell(m_T)$7, analogous exponential bounds follow for generalized diagonals and periodic billiard paths (Fu, 2024).

The limitations are intrinsic rather than technical. Example 3.1 shows that the additive constant cannot be removed uniformly for all trajectories. Example 6.1 shows that allowing negative curvature destroys uniform lower bounds. Example 6.3 shows that dependence on the curvature gap cannot be removed. These examples rule out the idea that $L(T)=\ell(m_T)$8 should hold uniformly without geometric restrictions (Fu, 2024).

A probabilistic variant studies the shortest geodesic between two distinguished singularities on a random flat cone sphere. Writing $L(T)=\ell(m_T)$9 for that distance and $T$0, the distribution of $T$1 with respect to Thurston’s volume form satisfies a recurrence built from a truncation flow and a cutting/gluing decomposition along multiple shortest geodesics. The base case for three cone points is a delta mass, and in the symmetric four-cone case $T$2, $T$3,

$T$4

Equivalently, the small-length density behaves as $T$5 as $T$6 (Rukhovich, 2022).

3. Hyperbolic cone surfaces and marked length data

On hyperbolic cone surfaces, the phrase is used differently. A hyperbolic cone surface is a $T$7-manifold triangulated by hyperbolic triangles, with constant curvature $T$8 away from finitely many singular points, possibly with cusps or geodesic boundary. Cone points have cone angles $T$9, and the paper unifies cone points, cusps, and geodesic boundary components through a generalized boundary assignment

$X$0

In that setting, “cone length” is the scalar $X$1 attached to a cone point (Pan, 2017).

The marked length spectrum is defined on isotopy classes of non-peripheral simple closed curves. For $X$2, $X$3 is the length of the unique geodesic representative of $X$4, and

$X$5

The principal rigidity result is finite marked length spectral rigidity for non-exceptional surfaces: there exists a finite set $X$6 of cardinality at most $X$7 such that equality of $X$8 on $X$9 determines the hyperbolic cone structure up to isotopy (Pan, 2017).

The proof reconstructs Fenchel–Nielsen data $x_1,\dots,x_n$0 from finitely many non-peripheral geodesic lengths. Boundary assignments across generalized $x_1,\dots,x_n$1-pieces are recovered from explicit hyperbolic formulas, and twist parameters are reconstructed from Dehn-twist length identities. One sample formula on a torus with one cone point is

$x_1,\dots,x_n$2

This makes the cone angle itself recoverable from ordinary geodesic length data (Pan, 2017).

The same framework supports a version of Thurston’s asymmetric metric on $x_1,\dots,x_n$3,

$x_1,\dots,x_n$4

which is shown to be well-defined, proper, and geodesic for non-exceptional surfaces with fixed boundary assignments. The resulting Teichmüller space is almost isometric to the punctured Teichmüller space $x_1,\dots,x_n$5, with constants $x_1,\dots,x_n$6 and $x_1,\dots,x_n$7 satisfying $x_1,\dots,x_n$8 and $x_1,\dots,x_n$9 as $x_i$0 (Pan, 2017).

The terminological point is precise: in this literature, cone length is not the length of a geodesic on a cone metric but a signed boundary parameter encoding cone angle.

4. Rational homotopy and the Lemaire–Sigrist problem

In rational homotopy theory, cone length is a homotopy invariant. For a path-connected space $x_i$1, $x_i$2 is the least integer $x_i$3 such that $x_i$4 is covered by $x_i$5 open sets contractible in $x_i$6. The cone length $x_i$7 is the minimum number of cofibrations needed to build a space of the homotopy type of $x_i$8 from a suspension by attaching suspensions. Formally, $x_i$9 if $2\pi(1-k_i)$0 is contractible; otherwise it is the smallest $2\pi(1-k_i)$1 for which there exist cofibration sequences

$2\pi(1-k_i)$2

with $2\pi(1-k_i)$3 and $2\pi(1-k_i)$4 (Parent et al., 14 Oct 2025).

For path-connected normal ANRs,

$2\pi(1-k_i)$5

Thus cone length differs from LS-category by at most one. This gave rise to the Lemaire–Sigrist conjecture that $2\pi(1-k_i)$6 for rational spaces. The conjecture is true for spaces of LS-category $2\pi(1-k_i)$7, and Félix–Thomas verified it for LS-category $2\pi(1-k_i)$8, but Dupont produced a rational counterexample with $2\pi(1-k_i)$9 and $k_i$0 (Parent et al., 14 Oct 2025).

The 2025 paper extends Dupont’s construction to every $k_i$1: for each such $k_i$2, it constructs a rational space $k_i$3 satisfying

$k_i$4

The construction is carried out in Quillen’s differential graded Lie algebra framework. A free dgl $k_i$5 has a decomposition of length $k_i$6 if

$k_i$7

for $k_i$8, with $k_i$9. The cone length of the dgl is the least such $\lambda(\Delta)=-\theta$00 up to quasi-isomorphism, and this equals the cone length of the modeled rational space (Parent et al., 14 Oct 2025).

The obstruction to shortening the decomposition is encoded by explicit generators $\lambda(\Delta)=-\theta$01, with

$\lambda(\Delta)=-\theta$02

The argument shows that no quasi-isomorphism can eliminate the last filtration stage represented by $\lambda(\Delta)=-\theta$03. A standard but incorrect intuition is that rational cone length should always coincide with LS-category; the construction disproves that in all degrees $\lambda(\Delta)=-\theta$04 (Parent et al., 14 Oct 2025).

5. Coxeter groups, cone types, and automata

In Coxeter theory, cone length is attached to a cone type rather than to a space. For a Coxeter system $\lambda(\Delta)=-\theta$05 with word length $\lambda(\Delta)=-\theta$06, the cone type of $\lambda(\Delta)=-\theta$07 is

$\lambda(\Delta)=-\theta$08

Equivalently, for $\lambda(\Delta)=-\theta$09,

$\lambda(\Delta)=-\theta$10

The paper defines the minimal representative $\lambda(\Delta)=-\theta$11 of a cone type $\lambda(\Delta)=-\theta$12 as the unique element of minimal length satisfying $\lambda(\Delta)=-\theta$13, and the cone length of $\lambda(\Delta)=-\theta$14 by

$\lambda(\Delta)=-\theta$15

This makes cone length a canonical statistic of a cone type, not of an arbitrary representative (Parkinson et al., 2021).

The main theorem states that every cone type has a unique minimal length representative $\lambda(\Delta)=-\theta$16, and if $\lambda(\Delta)=-\theta$17 satisfies $\lambda(\Delta)=-\theta$18, then $\lambda(\Delta)=-\theta$19 is a suffix of $\lambda(\Delta)=-\theta$20. Equivalently, in the partition $\lambda(\Delta)=-\theta$21, there is a unique minimal-length gate $\lambda(\Delta)=-\theta$22, and $\lambda(\Delta)=-\theta$23 (Parkinson et al., 2021).

The structural explanation uses regular partitions of $\lambda(\Delta)=-\theta$24. A partition is regular if it is locally constant on left descent sets and stable under left multiplication by simple reflections outside the common descent set. Regular partitions are essentially equivalent to automata recognizing the language of reduced words. The cone type partition is the minimal, or Myhill–Nerode, automaton for reduced words, and the gatedness of its parts yields the unique minimal representative (Parkinson et al., 2021).

Cone types also admit a root-theoretic description. If $\lambda(\Delta)=-\theta$25, then

$\lambda(\Delta)=-\theta$26

The paper identifies the minimal set of roots needed in such a description: the boundary roots

$\lambda(\Delta)=-\theta$27

and proves the minimal half-space representation

$\lambda(\Delta)=-\theta$28

This root-minimality is closely related to Brink–Howlett elementary inversion sets and Dyer’s base of an inversion set (Parkinson et al., 2021).

In finite Coxeter groups, cone types are in bijection with group elements: $\lambda(\Delta)=-\theta$29 so $\lambda(\Delta)=-\theta$30 and $\lambda(\Delta)=-\theta$31. In affine types, the result is presented as an analogue of Shi’s theorem that each region of the Shi arrangement has a unique minimal element (Parkinson et al., 2021).

6. Cone-associated length parameters in PDE, curve flow, and electromagnetics

Several papers use the phrase more broadly for a length attached to a cone-shaped geometry. This suggests a broader terminological pattern: the relevant length may be that of a generating curve, an evolving arc inside a cone, or the physical size of a conical device.

For a two-dimensional cone $\lambda(\Delta)=-\theta$32 generated by a smooth simple closed curve $\lambda(\Delta)=-\theta$33, the decisive parameter is the spherical length

$\lambda(\Delta)=-\theta$34

The free-boundary paper proves the sharp threshold

$\lambda(\Delta)=-\theta$35

For axisymmetric cones generated by a circle of latitude at polar angle $\lambda(\Delta)=-\theta$36,

$\lambda(\Delta)=-\theta$37

When $\lambda(\Delta)=-\theta$38, explicit examples even allow more than one positive phase to meet at the vertex (Allen et al., 2013).

For fourth-order curve diffusion flows inside planar cones, the evolving open curve has length $\lambda(\Delta)=-\theta$39 and satisfies one of three PDEs: $\lambda(\Delta)=-\theta$40 In the penalized case,

$\lambda(\Delta)=-\theta$41

so length decreases and extinction occurs in finite time. In the constrained cases, $\lambda(\Delta)=-\theta$42 or $\lambda(\Delta)=-\theta$43 is chosen so that $\lambda(\Delta)=-\theta$44 exactly. If the cone angle is less than $\lambda(\Delta)=-\theta$45, the initial curve has small oscillation of curvature, and the curve is sufficiently far from the cone tip, then the length-constrained flows exist for all time and converge exponentially in $\lambda(\Delta)=-\theta$46 to the unique circular arc centered at the tip with the same length $\lambda(\Delta)=-\theta$47. For a limiting circular arc of radius $\lambda(\Delta)=-\theta$48 in a cone of aperture $\lambda(\Delta)=-\theta$49,

$\lambda(\Delta)=-\theta$50

A similar statement holds for the penalized flow after rescaling (Gazwani et al., 14 Jun 2026).

In electromagnetics, a symmetric coaxial biconical antenna of half-angle $\lambda(\Delta)=-\theta$51 is capped at radius $\lambda(\Delta)=-\theta$52, and the paper explicitly calls $\lambda(\Delta)=-\theta$53 the antenna “length”; the physical tip-to-tip length is $\lambda(\Delta)=-\theta$54. The one-way transit time is

$\lambda(\Delta)=-\theta$55

and the time-domain receive effective length is

$\lambda(\Delta)=-\theta$56

Because the modal exponentials are normalized by $\lambda(\Delta)=-\theta$57, longer $\lambda(\Delta)=-\theta$58 produces slower decays and lower oscillation rates in physical time, and reflections appear at delays governed by $\lambda(\Delta)=-\theta$59 (Janaswamy, 2023).

7. Sum-of-squares length and adjacent algebraic usages

In the geometry of the SOS cone $\lambda(\Delta)=-\theta$60, cone length is the SOS length

$\lambda(\Delta)=-\theta$61

If

$\lambda(\Delta)=-\theta$62

with $\lambda(\Delta)=-\theta$63, then $\lambda(\Delta)=-\theta$64 equals the minimal rank of $\lambda(\Delta)=-\theta$65 over the Gram spectrahedron

$\lambda(\Delta)=-\theta$66

Thus cone length is simultaneously a decomposition length and a Gram-rank invariant (Laplagne et al., 2020).

The paper focuses on strictly positive polynomials on the boundary of the SOS cone, meaning $\lambda(\Delta)=-\theta$67 but every Gram matrix in $\lambda(\Delta)=-\theta$68 is singular. In the two classical Blekherman cases, the boundary length is rigid: if $\lambda(\Delta)=-\theta$69 and $\lambda(\Delta)=-\theta$70, then $\lambda(\Delta)=-\theta$71; if $\lambda(\Delta)=-\theta$72 and $\lambda(\Delta)=-\theta$73, then $\lambda(\Delta)=-\theta$74. In both cases the Gram matrix is essentially unique (Laplagne et al., 2020).

For general $\lambda(\Delta)=-\theta$75 and $\lambda(\Delta)=-\theta$76, the maximum possible length of strictly positive boundary points is governed by the Hankel index $\lambda(\Delta)=-\theta$77, giving

$\lambda(\Delta)=-\theta$78

The paper proves that the known bounds are optimal for all degrees and numbers of variables by constructing explicit strictly positive boundary polynomials attaining them. It also produces two types of counterintuitive examples: strictly positive boundary points with $\lambda(\Delta)=-\theta$79 and common complex roots among the summands, and boundary points whose length exceeds what naive dimension heuristics would suggest (Laplagne et al., 2020).

A final terminological caution comes from commutative algebra. The paper on Betti tables over three non-collinear points describes the cone of Betti tables of all finitely generated graded modules and, separately, the cone of Betti tables of all finite length modules. That cone is rational polyhedral and generated by pure diagrams, but the phrase concerns a cone indexed by finite length modules rather than an invariant called cone length (Gheorghita et al., 2014).

Taken together, these usages show that “cone length” is best treated as a family resemblance term. In metric cone geometry it measures geodesic size; in hyperbolic Teichmüller theory it encodes cone-angle data; in rational homotopy it counts cofibration stages; in Coxeter theory it records canonical minimal word length; and in SOS geometry it measures decomposition rank. The only safe global definition is local: the term acquires its precise meaning from the cone structure under discussion.