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On families of Finsler metrics

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Pith's one-line read Under a bi-geodesic condition, arithmetic symmetrisation of any Finsler metric is again Finsler, with the weighted norm as Lagrangian.

desk verdict Solid general theorems on symmetrizing Finsler metrics, but the advertised Funk-metric application is spoiled by a genuine degeneracy issue on unbounded domains. read the letter →

arxiv 2506.03749 v1 pith:PTNE2P2B submitted 2025-06-04 math.DG math.GTmath.MG

Ismail Saglam , Ken'Ichi Ohshika , Athanase Papadopoulos (IRMA) This is my paper
classification math.DGmath.GTmath.MG MSC 53C7051K0551K1053B4053C60
keywords SymmetrisationofaFinslermetricHilbertweightedFunkfamiliesmetricscompletenon-symmetricTeichmüllerspacespacesEuclideansurfacesThurston
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

A Finsler metric is a continuous family of possibly asymmetric norms on the tangent spaces of a manifold, and it induces a distance that need not be symmetric. This paper asks when weighted combinations of such a distance with its reverse, namely $d(F)^a_t(x,y)=(1-t)d(F)(x,y)+t d(F)(y,x)$ and the max version, are again induced by a Finsler norm. The central answer is that the arithmetic combination is Finsler whenever every pair of points is joined by a bi-geodesic, a path that is length-minimizing in both directions simultaneously, and the new Lagrangian is simply $F^a_t(x,v)=(1-t)F(x,v)+tF(x,-v)$. In the Funk geometry of convex domains, straight segments are bi-geodesics, so the weighted arithmetic Funk metrics are Finsler with straight-line geodesics, giving a one-parameter family of solutions to Hilbert's fourth problem. The same framework yields criteria for weighted sums and maxima of two Finsler metrics to be Finsler and applies to asymmetric metrics on Teichmüller spaces.

What carries the argument

The load-bearing object is the bi-geodesic: an absolutely continuous path $\gamma$ with $\ell_F(\gamma)=d(F)(x,y)$ and $\ell_F(\gamma^{-1})=d(F)(y,x)$. The proof of Theorem 5.3 lets this single path attain the two infima in Lemma 3.7 simultaneously, converting the general inequality $\delta(\ell(F^a_t))\ge d(F)^a_t$ into equality. The companion machinery is the identification of variational length with integral Finsler length for absolutely continuous paths (Lemma 4.7) and the identity $(\ell_F)_t=\ell_{F^a_t}$ (Corollary 4.20), which pass the arithmetic combination through the length-to-distance construction. For the max family, the controlling condition is a sign condition on the difference $(1-t)F_1-tF_2$ along a common geodesic.

What would settle it

Integrate the Lagrangian $p^a_t$ given in Section 6 for the upper half-space model, namely $p^a_t(x,v)=t v_n/x_n$ for $v_n>0$ and $(1-t)|v_n|/x_n$ for $v_n<0$, along the straight segment between two points and compare with the weighted Funk distance $F^a_t$; any $t\in(0,1)$ for which the two numbers differ would refute Corollary 6.5 and the bi-geodesic transfer principle behind it.

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Extended reading notes

Core claim

The core discovery is that reversibility of geodesics, not symmetry of the norm, is the controlling condition for whether symmetrised distances stay Finsler. Theorem 5.3 states that if a Finsler structure $F$ on a smooth manifold admits an absolutely continuous bi-geodesic between every pair of points, then for every $t\in[0,1]$ the arithmetic weighted distance $d(F)^a_t$ is induced by the Finsler structure $F^a_t(x,v)=(1-t)F(x,v)+tF(x,-v)$, and every bi-geodesic of $F$ is also a bi-geodesic of $F^a_t$. Because straight segments are bi-geodesics for the Funk metric on a convex domain, Corollary 6.5 makes the weighted arithmetic Funk metrics Finsler with straight-line geodesics. The paper further shows that a weighted sum $(1-t)F_1+tF_2$ induces the weighted distance exactly when a common absolutely continuous geodesic exists, and that the max version requires, in the smooth complete case, a common geodesic along which one Lagrangian dominates the other.

Load-bearing premise

The load-bearing premise is that every pair of points can be joined by a path that is shortest in both directions at once, which Theorem 5.3 requires and Section 7 does not verify for the Thurston-like metrics it discusses.

Editorial extensions

If this is right

  • For any convex domain, the weighted arithmetic Funk metrics $F^a_t$ are Finsler for every $t$, and the straight Euclidean segments are their geodesics, so this is a family of solutions to Hilbert's fourth problem.
  • The Hilbert metric is the $t=1/2$ member of this family, and its Finsler Lagrangian is recovered as the arithmetic symmetrisation of the Funk Lagrangian, now extended from $t=1/2$ to all weights.
  • Any Finsler manifold satisfying the bi-geodesic hypothesis inherits a geodesic-sharing property: the same paths that are bi-geodesics for $F$ are bi-geodesics for $F^a_t$.
  • Sums and maxima of complete Finsler structures are complete, and forward and backward convergence always agree for Finsler-induced distances, which reproduces the convergence behaviour known for Thurston's and the earthquake metrics.
  • For $t\neq 0$, the weighted arithmetic Funk metric is uniquely geodesic precisely when the boundary contains no pair of non-collinear Euclidean segments in a common plane, matching the Hilbert-metric criterion.

Reading between the lines

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

  • The bi-geodesic hypothesis suggests a concrete check for other asymmetric metrics: if some pair lacks a two-way shortest path, the arithmetic symmetrisation could still be Finsler, but the equality in Theorem 5.3 would not follow from this argument; Section 7's Teichmüller metrics are the natural place to test whether the condition is necessary.
  • The sign condition in the max theorem looks like a no-overtaking condition along a common geodesic; on convex domains, testing it for the weighted max Funk metrics would explain exactly where the counterexample in Remark 6.3 enters and whether other weights behave better.
  • Using the explicit Lagrangian in the ball and upper half-space models, one could compute flag curvature or projectivity of the weighted Funk metrics; the paper establishes geodesics and Finslerity but leaves those derived quantities open.
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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

1 major / 6 minor

Summary. The paper studies two one-parameter families of metrics derived from a Finsler metric F: the arithmetic family d(F)^a_t = (1-t)d(F)+t d(F)^{-1} and the max family d(F)^m_t = max{(1-t)d(F), t d(F)^{-1}}. It gives conditions under which these distance functions are induced by Finsler structures with Lagrangians F^a_t = (1-t)F+tF(x,-v) and F^m_t = max{(1-t)F, tF(x,-v)}. The main positive results are Theorem 5.1 (characterization via common geodesics for two Finsler metrics), Theorem 5.3 (bi-geodesic hypothesis implies the arithmetic symmetrization is Finsler), Theorem 5.4 (sign condition for the max Lagrangian), and a necessary-and-sufficient condition in Theorem 5.5. The paper also develops weak-metric tools (completeness, Arzelà–Ascoli, Hopf–Rinow in the asymmetric setting) and applies the results to Funk and Hilbert geometries, where the arithmetic weighted Funk metrics are shown to have straight-line geodesics. Section 7 formulates open questions for Thurston-like metrics on Teichmüller spaces.

Significance. If the main theorems are correct, they provide a clean synthesis of known symmetrization phenomena in Finsler and Funk/Hilbert geometry, with machine-checkable-level detail in the proofs of Theorems 5.1, 5.3, and 5.4. The explicit use of absolute continuity and bi-geodesics is a genuine improvement over earlier heuristic treatments. The application to weighted Funk metrics and the connection to Hilbert's fourth problem are attractive. However, the advertised application in Corollary 6.5 is not valid for unbounded convex domains, and the necessity proof of Theorem 5.5 has a gap; these need to be resolved before the results can be accepted in full.

major comments (1)
  1. [Section 6, Corollary 6.5 and the upper half-space example] The statement of Corollary 6.5 is false as written for unbounded convex sets. For the upper half-space H={x_n>0}, the Funk Lagrangian p(x,v)=inf{t>0 | x+v/t∈H} vanishes whenever v_n≥0, including all horizontal vectors v with v_n=0. Consequently p^a_t(x,v)=(1-t)p(x,v)+tp(x,-v) also vanishes on horizontal vectors, contradicting Definition 4.1(3), which requires F(x,v)=0 if and only if v=0. The induced distance satisfies F^a_t(x,y)=0 for two distinct points on the same horizontal level, so p^a_t is not a Finsler structure in the sense of the paper. The displayed formula p(x,v)=max(v_n/x_n,0) is also incorrect: the correct expression is p(x,v)=max(-v_n/x_n,0), and the subsequent piecewise formula for p^a_t is consistent with this corrected p. The corollary can be repaired by restricting to bounded convex sets (or more generally to convex sets containing no affine line), but this restriction is not stated anywhere in Section 6; indeed the section explicitly allows arbitrary closed convex sets and presents the upper half-space as an example. This affects the advertised application to Funk geometry and the Hilbert-fourth-problem interpretation in Proposition 6.1.
minor comments (6)
  1. [Section 2, Example 2.1] The notation 'length dm (α[a,t0])' is slightly ambiguous; it would be clearer to write 'the length of the subarc α|[a,t0] with respect to the norm ‖·‖_m'.
  2. [Section 4, Definition 4.14 and Lemma 4.15] The metric d(F) is denoted dF in Definition 4.14 and Lemma 4.15 without being defined; please use d(F) consistently throughout.
  3. [Section 4, proof of Proposition 4.13] The sentence 'She other inequality' contains a typo; it should be 'The other inequality'.
  4. [Section 5, Theorem 5.5 proof] There are several typos in the displayed formulas: the final line writes 'd(max{(1-t)F1,F2})' where tF2 is meant, and earlier 'max{td(F1),(1-t)d(F2)}' swaps the arguments. These make the case analysis harder to follow.
  5. [Section 6, first paragraph] The section begins by taking Ω to be a closed convex set, but Corollary 6.5 refers to 'an open convex set'; the relationship between these two conventions (and the domain on which the Funk metric is defined) should be clarified.
  6. [Section 1, outline] The outline says Busemann's axiom is introduced in Section 4, but it is actually defined in Definition 3.2 in Section 3; please correct the cross-reference.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: weighted Finsler conclusions follow from explicit bi-geodesic hypotheses, not from definitional identifications.

full rationale

The central derivation chain is self-contained. Theorem 5.3's equality δ(l(F^a_t)) = d(F)^a_t is not a definitional identity: the lower bound in Eq. (11) is the always-true inf-over-sum inequality, and the matching upper bound uses the genuinely substantive hypothesis that every pair of points is joined by an absolutely continuous bi-geodesic. Corollary 6.5 applies this theorem to Funk geometry, where straight segments serve as bi-geodesics; the corollary's conclusion is the theorem's conditional conclusion, not a restatement of the definitions of F^a_t or p^a_t. Theorems 5.1, 5.4, and 5.5 similarly reduce to their explicit hypotheses (common geodesics, sign conditions) and are proven from Lemma 4.7 and Corollary 4.20. Self-citations ([16], [17], [26], [32]) are contextual, historical, or used for comparison; none is load-bearing, and [26] is explicitly generalized with a proof in the text. No fitted parameter is renamed as a prediction. One non-circular correctness concern should be noted separately: in the half-space example of Section 6, the displayed p(x,v)=max(v_n/x_n,0) yields p(x,v)=0 for horizontal vectors, contradicting Definition 4.1(3); this affects the validity of Corollary 6.5 for unbounded Ω, but it is an error in applying Theorem 5.3, not a circular reduction of the paper's claims.

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

No free parameters or fitted constants appear; the paper is purely analytic. The 'axioms' are the paper's definitions, standing hypotheses, and imported standard theorems. The weighted families F^a_t and F^m_t are mathematical objects defined for study, not ad hoc postulated entities, so invented_entities is empty.

assumptions (6)
  • domain assumption Finsler structures are continuous weak norms on tangent spaces (Definition 4.1), with no symmetry, smoothness, or strong convexity required.
    All later theorems are stated in this generality; the results depend on this weak definition, e.g., in Proposition 4.4 and the Hopf-Rinow variant.
  • domain assumption All weak metric spaces are assumed to satisfy Busemann's axiom (Section 3).
    The paper states 'we shall assume that the spaces we consider satisfy Busemann's axiom'; Proposition 4.4 verifies it for Finsler metrics, while the general framework takes it as a standing hypothesis.
  • domain assumption Existence of absolutely continuous geodesics or bi-geodesics between every pair of points (hypotheses of Theorems 5.1, 5.3, 5.4 and Corollary 6.5).
    These are explicit conditions; e.g., Theorem 5.3 assumes a bi-geodesic for every x,y, and Corollary 6.5 inherits it from straight segments in convex domains.
  • domain assumption For Theorem 5.5, the two Finsler structures are complete and C∞.
    The necessity proof invokes Proposition 4.18, which requires C∞ regularity and completeness.
  • standard math Arzelà-Ascoli theorem for asymmetric metric spaces (Collins-Zimmer [8]).
    Used to prove Proposition 4.13 (Hopf-Rinow) and used in Example 2.3.
  • standard math Complete C∞ Finsler manifolds admit C∞ geodesics (Bao-Chern-Shen, Theorem 6.6.1 [2]).
    Imported as Proposition 4.18 and used in Theorem 5.5 necessity.

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Pith. "Pith review of On families of Finsler metrics." pith.science (2026). https://pith.science/paper/PTNE2P2B

@misc{pith2026250603749,
  author       = {Pith},
  title        = {Pith review of: On families of Finsler metrics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PTNE2P2B}},
  note         = {Machine review of arXiv:2506.03749}
}
read the original abstract

In this paper, we answer some natural questions on symmetrisation and more general combinations of Finsler metrics, with a view towards applications to Funk and Hilbert geometries and to metrics on Teichm{\"u}ller spaces. For a general non-symmetric Finsler metric on a smooth manifold, we introduce two different families of metrics, containing as special cases the arithmetic and the max symmetrisations respectively of the distance functions associated with these Finsler metrics. We are interested in various natural questions concerning metrics in such a family, regarding its geodesics, its completeness, conditions under which such a metric is Finsler, the shape of its unit ball in the case where it is Finsler, etc. We address such questions in particular in the setting of Funk and Hilbert geometries, and in that of the Teichm{\"u}ller spaces of several kinds of surfaces, equipped with Thurstonlike asymmetric metrics.

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