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arxiv: 2507.10067 · v2 · submitted 2025-07-14 · 🧮 math.MG

The Maximum of the Volume of a Cevian Simplex and its Parts

Yagub N. Aliyev This is my paper

Pith reviewed 2026-05-19 05:11 UTC · model grok-4.3

classification 🧮 math.MG
keywords cevian simplexvolume ratiobarycentric coordinatessimplexcentroidoptimization
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The pith

The volume of the cevian simplex inside an n-simplex is at most 1/n^n of the total, achieved when the interior point is the centroid.

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

The paper generalizes a known two-dimensional result to higher dimensions. In a triangle the cevian triangle formed by lines from an interior point to the vertices has area at most one quarter of the whole triangle, with the maximum when the point is the centroid. The same construction in an n-dimensional simplex yields a cevian simplex whose volume is at most 1 over n to the power n of the original volume. Barycentric coordinates supply the explicit volume ratios that prove the bound and also settle two optimization questions about the smaller simplices created by the cevians.

Core claim

For any interior point M of an n-simplex the volume of the cevian simplex is at most n^{-n} times the volume of the original simplex, with equality precisely when M is the barycenter; the same coordinate expressions determine the volumes of the sub-simplices and thereby solve the stated optimization problems.

What carries the argument

Barycentric coordinates of the interior point, whose products give the volume ratios of all the smaller simplices created by the cevians.

If this is right

  • The bound 1/n^n holds uniformly for every n and every simplex.
  • Equality occurs only when the interior point is the centroid.
  • The same ratios resolve two concrete optimization problems on the volumes of the divided parts.

Where Pith is reading between the lines

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

  • The result suggests that centroid-based partitions maximize certain volume functionals in simplicial domains.
  • The coordinate method could be tested on weighted volume ratios or on simplices with additional interior constraints.
  • Similar ratio calculations might apply to volume optimization inside polytopes that can be triangulated.

Load-bearing premise

The volume ratios obtained from barycentric coordinates in the plane extend unchanged to n dimensions for every interior point and every simplex shape.

What would settle it

Pick a regular tetrahedron, place a point at its centroid, compute the volume of the resulting cevian tetrahedron, and check whether the ratio equals exactly 1/27.

read the original abstract

The cevian triangle corresponding to an interior point $M$ of a triangle is the triangle determined by the feet of the three cevians concurrent at $M$. It is known that the area of the cevian triangle for an interior point $M$ of a triangle is at most $\frac{1}{4}$ of the area of the triangle, with maximum attained when $M$ is the triangle's centroid. This can be generalized from triangles to $n$-dimensional simplices, with $\frac{1}{4}$ replaced by $\frac{1}{n^n}$, using barycentric coordinates. We also use this method to solve two optimization problems about the parts of this simplex.

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

0 major / 3 minor

Summary. The manuscript generalizes the known 2D result that the cevian triangle of an interior point M in a triangle has area at most 1/4 of the triangle (attained at the centroid) to n-dimensional simplices. It claims that the volume of the corresponding cevian simplex is at most 1/n^n of the simplex volume, with the bound attained at the centroid, and derives this via barycentric coordinates. The same method is applied to solve two optimization problems on the volumes of the parts of the simplex.

Significance. If the central claim holds, the work supplies a clean, explicit, and parameter-free upper bound on the volume ratio that holds for arbitrary interior points and any simplex shape. The barycentric derivation identifies a unique interior critical point at the centroid by analyzing the function with gradient condition and boundary behavior, which is a methodological strength. Solving the two optimization problems extends the utility of the approach in geometric optimization.

minor comments (3)
  1. The abstract refers to 'two optimization problems about the parts of this simplex' without stating them explicitly; these should be formulated in the introduction or a dedicated section for clarity.
  2. The definition of the cevian simplex in n dimensions (via feet of cevians to the faces) should be stated formally with notation for the vertices and the matrix B whose determinant gives the volume ratio.
  3. Include a brief verification that the volume ratio expression evaluates to exactly 1/n^n when all barycentric coordinates a_i equal 1/(n+1).

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive evaluation of the manuscript, including recognition of the clean upper bound 1/n^n and the utility of the barycentric approach for the two optimization problems. The recommendation of minor revision is noted. No specific major comments appear in the report, so we have no individual points requiring direct response or manuscript changes at this stage.

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper derives the volume ratio of the cevian simplex directly from barycentric coordinate expressions, producing the explicit formula |det(B)| = n ⋅ (∏ a_i) ⋅ (∏ 1/(1−a_k)). It then identifies the maximum by solving the critical-point equations ∇log f = λ∇(∑a_i) under the simplex constraint ∑a_i=1, confirming the unique interior solution at a_i=1/(n+1) and showing the ratio approaches zero on the boundary. This chain uses only standard properties of determinants and constrained optimization; no step redefines an output as an input, renames a fitted quantity as a prediction, or relies on a load-bearing self-citation whose validity depends on the present work.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard properties of barycentric coordinates and volume scaling in simplices; no free parameters or invented entities are introduced.

axioms (1)
  • standard math Barycentric coordinates express any interior point as convex combination of vertices and allow direct computation of sub-simplex volumes.
    Invoked to generalize the 2D area ratio to n dimensions and to set up the optimization problems.

pith-pipeline@v0.9.0 · 5640 in / 1314 out tokens · 29146 ms · 2026-05-19T05:11:32.085548+00:00 · methodology

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Reference graph

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15 extracted references · 15 canonical work pages

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