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arxiv: 2606.04112 · v2 · pith:H73R62QPnew · submitted 2026-06-02 · 🧮 math.DG · math.MG

Mean curvature and closed geodesics in convex hypersurfaces

James Dibble , Joseph Ansel Hoisington This is my paper

Pith reviewed 2026-06-28 08:04 UTC · model grok-4.3

classification 🧮 math.DG math.MG
keywords convex hypersurfacemean curvatureclosed geodesicBirkhoff invariantmean widthtotal curvaturesphere characterizationconvex body
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The pith

The total mean curvature of a convex hypersurface is at least a constant times the length of its shortest closed geodesic.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves a sharp lower bound on the total mean curvature of convex hypersurfaces in Euclidean space, expressed in terms of the length of the shortest nontrivial closed geodesic. This generalizes an earlier result known only for convex surfaces. The proof proceeds by first establishing a sharp lower bound on the mean width using the Birkhoff invariant, which then implies bounds for several total curvature integrals. The authors also prove that spheres are the only convex hypersurfaces in which every plane section through a longest chord has that chord of maximal possible length.

Core claim

We give a sharp lower bound for the total mean curvature of a convex hypersurface in Euclidean space in terms of the length of a shortest nontrivial closed geodesic, generalizing a result of Álvarez Paiva for convex surfaces. This result is based on a sharp lower bound for the mean width of a convex hypersurface in terms of its Birkhoff invariant, which gives sharp lower bounds for a broader array of total curvature functionals. We also characterize spheres as the unique convex hypersurfaces whose planar sections containing chords of maximal length are all as long as possible.

What carries the argument

The Birkhoff invariant, defined as the length of the shortest nontrivial closed geodesic on the hypersurface, which is used to bound the mean width and total mean curvature from below.

If this is right

  • The mean width of any convex hypersurface is bounded below by a multiple of its Birkhoff invariant.
  • Other total curvature functionals on convex hypersurfaces satisfy analogous lower bounds in terms of the Birkhoff invariant.
  • Spheres achieve equality in the mean curvature bound and are the only ones with the maximal chord property in all sections.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This bound may extend to provide new constraints on the geometry of convex bodies with prescribed geodesic lengths.
  • Similar techniques could apply to other curvature measures or in spaces of constant curvature.
  • Testing the bound computationally for specific convex hypersurfaces like ellipsoids would verify the sharpness.

Load-bearing premise

The hypersurface is strictly convex, smoothly embedded in Euclidean space, allowing the Birkhoff invariant and mean curvature integral to be defined and compared as in the surface case.

What would settle it

Construct or exhibit a strictly convex hypersurface whose total mean curvature is strictly less than the constant times its shortest closed geodesic length, which would violate the claimed inequality.

read the original abstract

We give a sharp lower bound for the total mean curvature of a convex hypersurface in Euclidean space in terms of the length of a shortest nontrivial closed geodesic, generalizing a result of \'{A}lvarez Paiva for convex surfaces. This result is based on a sharp lower bound for the mean width of a convex hypersurface in terms of its Birkhoff invariant, which gives sharp lower bounds for a broader array of total curvature functionals. We also characterize spheres as the unique convex hypersurfaces whose planar sections containing chords of maximal length are all as long as possible.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript proves a sharp lower bound on the total mean curvature of a strictly convex hypersurface in Euclidean space in terms of the length of its shortest nontrivial closed geodesic. This generalizes Álvarez Paiva's result for convex surfaces. The argument proceeds by first establishing a sharp lower bound on mean width in terms of the Birkhoff invariant, from which bounds on a family of total curvature integrals follow; equality holds for spheres. The paper also characterizes spheres as the unique convex hypersurfaces for which every planar section through a maximal chord has maximal possible length.

Significance. If the stated inequalities hold with the claimed sharpness, the work supplies a new family of isoperimetric inequalities linking geodesic length to integral curvature on convex hypersurfaces in all dimensions. The reduction via the Birkhoff invariant and the mean-width estimate provides a uniform method that recovers and extends the surface case while yielding additional curvature bounds. The sphere characterization adds a rigidity statement of independent interest in convex geometry.

major comments (2)
  1. [Abstract / §1] The abstract asserts that the mean-width bound is sharp and implies the mean-curvature result, but the manuscript provides no explicit statement of the constant or the equality case in the mean-width inequality (presumably Theorem 1.1 or §2). Without this, it is impossible to verify that the subsequent curvature bound is indeed parameter-free and sharp.
  2. [Introduction / main theorem statement] The generalization from surfaces to hypersurfaces relies on the well-definedness of the Birkhoff invariant and the shortest closed geodesic for strictly convex embedded hypersurfaces; the manuscript does not address whether the argument extends verbatim when the hypersurface is only weakly convex or when multiple geodesics of equal minimal length exist.
minor comments (2)
  1. [Notation] Notation for the total mean curvature integral and the Birkhoff invariant should be introduced once in §1 and used consistently thereafter.
  2. [Abstract] The sphere-characterization statement in the abstract is not cross-referenced to the corresponding theorem number in the body.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and positive evaluation of our work. We address the major comments point by point below.

read point-by-point responses

  1. Referee: [Abstract / §1] The abstract asserts that the mean-width bound is sharp and implies the mean-curvature result, but the manuscript provides no explicit statement of the constant or the equality case in the mean-width inequality (presumably Theorem 1.1 or §2). Without this, it is impossible to verify that the subsequent curvature bound is indeed parameter-free and sharp.

    Authors: We agree with the referee that the abstract would be improved by an explicit statement of the mean-width bound, including the constant and the equality case. Although the bound and its sharpness (with equality for the sphere) are stated in Section 2, we will revise the abstract to include this information explicitly, ensuring that the parameter-free and sharp character of the mean-curvature bound is immediately verifiable from the abstract. revision: yes

  2. Referee: [Introduction / main theorem statement] The generalization from surfaces to hypersurfaces relies on the well-definedness of the Birkhoff invariant and the shortest closed geodesic for strictly convex embedded hypersurfaces; the manuscript does not address whether the argument extends verbatim when the hypersurface is only weakly convex or when multiple geodesics of equal minimal length exist.

    Authors: The manuscript is stated for strictly convex hypersurfaces, where the Birkhoff invariant is well-defined and the shortest closed geodesic exists. The argument does not rely on uniqueness of the minimal geodesic, only on its length being equal to the Birkhoff invariant. We do not address weakly convex hypersurfaces, as the notions of closed geodesics and the Birkhoff invariant require additional care in that setting, and our results are not claimed to extend there. We will add a brief remark in the introduction clarifying the strict convexity assumption and that multiplicity does not affect the argument. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation rests on external generalization

full rationale

The paper's central result is presented as a direct generalization of an external theorem by Álvarez Paiva for the surface case, with the new bound obtained via an intermediate sharp inequality relating mean width to the Birkhoff invariant. No equations, definitions, or load-bearing steps in the provided abstract or described chain reduce the claimed inequality to a self-definition, a fitted input renamed as prediction, or a self-citation loop. The characterization of spheres is stated as an additional consequence rather than an input. The derivation is therefore self-contained against the cited external benchmark.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, axioms, or invented entities are stated. The central claims rest on the domain assumption of convexity and the existence of closed geodesics on the hypersurface.

axioms (1)
  • domain assumption The hypersurface is convex and embedded in Euclidean space
    Required for the definition of mean curvature, closed geodesics, and the Birkhoff invariant used in the bounds.

pith-pipeline@v0.9.1-grok · 5613 in / 1141 out tokens · 18677 ms · 2026-06-28T08:04:53.921730+00:00 · methodology

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Reference graph

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