A Generalization of a Classical Geometric Extremum Problem

Nikolai Nikolov; Oleg Mushkarov; Petar Kenderov

REVIEW 2 major objections 4 minor 14 references

A Generalization of a Classical Geometric Extremum Problem

T0 review · 2 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read For a convex body, every locally shortest chord through an interior point has smooth endpoints and concurrent boundary normals.

desk verdict The core local-minimizer theorem is solid, but the abstract's longest-chord and exterior-O claims are stated without proof, so the preprint overclaims while still deserving a serious referee. read the letter →

arxiv 2506.07252 v1 pith:FSBFHLLA submitted 2025-06-08 math.MG math.OC

Petar Kenderov , Oleg Mushkarov , Nikolai Nikolov This is my paper

classification math.MGmath.OC MSC 51M1652A2052A4049K0549K10

keywords convexbodyPhilolinechordextremumconcurrentperpendicularssupportinghyperplanepolytopelocalminimizersmoothness

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

The paper generalizes the classical geometric extremum problem of finding the shortest chord through a fixed interior point of a convex figure to arbitrary compact convex bodies in $\mathbb{R}^n$. Its central claim is that every locally shortest chord through an interior point must meet the boundary at points where the body is smooth, and the normals to the unique supporting hyperplanes at those points meet on the hyperplane through the point perpendicular to the chord. For smooth bodies, the same concurrency condition holds for locally longest chords, and when the point lies outside the body, analogous statements hold for chords passing through the body's interior. The paper also derives dimension-counting restrictions on the endpoints of longest chords through polytopes, showing in particular that planar polygons force a vertex, while a tetrahedron example shows the count is sharp.

What carries the argument

The load-bearing object is the Concurrent Perpendiculars Property (CPP): for a chord $[AB]$ cut from a convex set by a line through $O$, the perpendiculars to the supporting lines at $A$ and $B$ and the perpendicular to $AB$ at $O$ meet at a single point, equivalently, the two boundary normals and the line through $O$ perpendicular to $AB$ are concurrent. The planar proofs rest on a derivative identity: if $\phi$ is the angle of a rotating line through $O$, then $d|OA|/d\phi = |OA|\tan\phi$, which makes the chord-length function for a wedge strictly convex, so a critical point is the unique minimum and corresponds exactly to CPP. A tangency lemma shows that the distance from $O$ to a smooth curve has the same derivative as the distance to its tangent line, transferring CPP from supporting lines to smooth boundaries. In higher dimensions, the argument slices the body by two-dimensional planes built from the intersection of the supporting hyperplanes and reduces the $n$-dimensional theorem to the planar statements.

What would settle it

Numerically enumerate all chords through an interior point of a convex polytope in $\mathbb{R}^3$ and locate the shortest one: if any endpoint lies on an edge or vertex, or if the two normals to the supporting planes plus the perpendicular through $O$ fail to meet in one point, Theorem 21 is false. A direct test of the slicing step is to take the tetrahedron example from Section 4, cut it by the plane through a candidate shortest chord and the intersection line of its supporting planes, and check whether that chord is still a local minimizer in the slice.

Watch

Extended reading notes

Core claim

On the paper's own terms, the discovery is that the ancient perpendicular-concurrence characterization of the shortest segment cut from an angle extends to all convex bodies in all dimensions, in local form. Theorem 21 states: if $O$ is an interior point of a closed convex set $\mathcal{C}$ in $\mathbb{R}^n$ and $[A^*B^*]$ is a local minimizer of the length $F([AB])$ among all chords through $O$, then there is a unique supporting hyperplane at $A^*$ and a unique supporting hyperplane at $B^*$, and the normal lines to those hyperplanes intersect at a point lying on the hyperplane through $O$ orthogonal to $A^*B^*$. The smooth-body counterpart (Theorem 22) and the exterior variants (Theorems 24 and 25) extend the same conclusion to local maxima and to points outside $\mathcal{C}$. The planar case is settled first, for angles and for general convex sets, and the higher-dimensional statements are obtained by slicing with two-dimensional planes.

Load-bearing premise

The proof's key step assumes that if a chord is a local minimizer in $\mathbb{R}^n$, it remains a local minimizer when the body is cut down to an arbitrary two-dimensional slice through the intersection of its supporting hyperplanes; if a slice admitted a shorter nearby chord, the concurrency conclusion would not follow from the argument given.

Editorial extensions

If this is right

Every locally shortest chord through an interior point has smooth endpoints, so a corner or edge of a convex body can never support a shortest chord through that point.
The concurrency condition gives a finite geometric check for candidate shortest chords: the boundary normals at the two endpoints must meet on the hyperplane through $O$ perpendicular to the chord.
For smooth convex bodies, the same concurrency condition characterizes both local maxima and local minima of chord length, and the chord length as a function of rotation angle is differentiable.
For polytopes, any chord that is a local maximum of $F$ through an interior point has endpoints whose face-dimensions sum to at most $n-1$; in the plane this forces at least one endpoint to be a vertex, and the tetrahedron example shows this bound cannot be improved in higher dimensions.
When $O$ lies far outside a polytope without parallel facets, the exterior length function $G$ has no local minimizers that pass through the body's interior.

Reading between the lines

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

The paper does not develop an algorithmic reading, but the concurrency condition could serve as a computational filter: instead of sampling all chords through $O$, one could search pairs of boundary points whose normals intersect on the perpendicular hyperplane, reducing the problem to a lower-dimensional root-finding task.
Because the planar length function is strictly convex, the slice argument suggests that on strictly convex bodies the shortest chord through $O$ is unique, a global uniqueness statement the paper does not explicitly make for general convex bodies.
The dimension-counting estimate for polytopes yields a practical face-pair filter that the paper leaves implicit: to find a longest chord through an interior point of a polytope, only pairs of faces whose dimensions sum to at most $n-1$ need to be searched.
A natural testable extension is to replace the supporting hyperplanes in the concurrency statement by normal cones at non-smooth points; the theorem forces uniqueness only at the optimizer, so a generalized concurrency statement might still hold there.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Here's my read on 2506.07252. The headline result—Theorem 21, the concurrent-perpendiculars property for local minimizers of chords through an interior point in R^n—is proved in detail and I think it is correct. The step that worried the reader, transferring local minimality from R^n down to a 2D slice, is actually fine: a relative boundary point of C∩Π is a boundary point of C, so the ambient local-minimality inequality applies verbatim in the slice. I don't see a gap there.

What is genuinely new is the n-dimensional necessary condition for arbitrary convex bodies, plus the polytope dimension bounds (Props 28, 30, 32). The 2D sections are rigorous and self-contained, and the historical tracing through Eves, Neovius, Lhuilier, and Coxeter–van de Craats is accurate and useful.

The real soft spot is the gap between abstract and proofs. The abstract announces the property for the longest chord of a smooth convex body and for exterior O. But Theorem 22 and Theorem 25 are stated with proofs omitted, and Lemma 18—the exterior analog of Lemma 13—is also asserted without proof. Since a global maximizer is a local maximizer, the longest-chord claim depends entirely on the omitted proof of Theorem 22. This isn't a minor omission; it's the headline generalization. Theorem 24 is also a sketch, following Theorem 21 'step-by-step.' The polytope arguments are mostly proven, though a few steps in Proposition 28 are terse; the local-max interplay with the angle function would benefit from a few extra sentences.

The citation pattern is clean, no circularity, no fitted constants. The self-citation [6] is only in the appendix and is irrelevant to the theorems. The paper is written by people who know this history and the geometry.

Bottom line: the core result is solid, reusable, and worth citing. But the preprint overclaims relative to what is actually derived. Send it to peer review—a good referee will ask for complete proofs of Theorems 22 and 25 and Lemma 18. With those supplied, this becomes a clean paper. I'd bring it to a reading group only if someone is actively working on chord problems; otherwise it's a wait-for-the-revision.

Referee Report

2 major / 4 minor

Summary. The paper studies the extremal problem of segments cut from a compact convex body by lines through a fixed point O, and proves necessary conditions of a 'Concurrent Perpendiculars Property' (CPP) type. In the plane, the authors prove CPP for local minima of the chord-length function when O is interior (Proposition 5), for local maxima when the body is smooth (Propositions 10 and 17), and for exterior-point variants (Propositions 16 and 17). In higher dimensions, Theorem 21 proves CPP and uniqueness of the supporting hyperplanes at the endpoints for local minimizers when O is interior, and the proof is given in detail. Theorems 22 and 25, which would give the corresponding local-maximizer results for smooth bodies, and Lemma 18, the exterior analogue of Lemma 13, are stated with proofs omitted. Section 4 gives dimension bounds for local maximizers of polytopes and two examples. I also checked the suspected gap in the proof of Theorem 21 concerning restriction to a two-dimensional slice; that step is valid, though the text should supply the one-line justification.

Significance. If fully proved, the paper would establish a clean higher-dimensional analogue of the classical Philo-line property, with a necessary optimality condition that is easy to test geometrically. The 2D lemmas in Section 2 are rigorous, and the proof of Theorem 21 is a genuine deductive construction with no fitted parameters or data-dependent assumptions. However, the abstract advertises the longest-chord result for smooth bodies and the exterior-point analogues, and those claims rest on omitted proofs. The contribution as submitted is therefore a solid local-minimizer theorem together with plausible but unproved generalizations; the latter need to be completed before the advertised results can be credited.

major comments (2)

[Section 3, Theorem 22] Theorem 22 is stated with 'The proof of this theorem is omitted.' Since a globally longest chord is, in particular, a local maximizer in the sense of Definition 20, the abstract's assertion that CPP holds for the longest segment of a smooth convex body depends entirely on this unproved theorem. The two sentences after the theorem describe a route via Lemma 13 and Proposition 10, but they do not supply a proof; moreover, Lemma 23, which is needed for this route, is also asserted without proof. A complete proof of Theorem 22 and of Lemma 23 is required before the corresponding abstract claim can be considered established.
[Section 2, Lemma 18; Section 3, Theorem 25] Lemma 18, the exterior analogue of Lemma 13, is stated with 'The proof of this statement is omitted because it is quite similar to the proof of Lemma 13.' Theorem 25, the exterior local-maximizer result advertised in the abstract, then cites this lemma as a load-bearing ingredient. The exterior configuration is not identical to the interior one: the objective is g(phi)=|OA|-|OB| rather than f(phi)=|OA|+|OB|, and the relative positions of O, A, and B differ. A full proof of Lemma 18 and a complete proof of Theorem 25 are needed; alternatively, the abstract and the theorems must be reworded to state explicitly that these are announced results rather than proved ones. The same applies to Theorem 24, whose proof is only summarized as following step-by-step from Theorem 21.

minor comments (4)

[Section 3, proof of Theorem 21] In the paragraph beginning 'Take an arbitrary non-zero vector h', the transference of local minimality from C∩Π(h) to the strip between l1(h) and l2(h) is asserted without proof. It is correct: because the strip contains C∩Π(h), every chord of C∩Π(h) is contained in the corresponding strip chord, so |strip chord| ≥ |chord in C∩Π(h)| ≥ |A*B*|. Please add this sentence. Also note that A*B* cannot be parallel to L, since otherwise the interior point O would lie in a supporting hyperplane, which justifies that l1(h) and l2(h) determine a genuine plane.
[Throughout] There are numerous typographical errors that should be corrected in revision, including 'hiperplane' in the abstract, 'boudary', 'considetations', 'the the tangents' in Proposition 17, 'segemnt' in Proposition 28, 'valide', 'dufferent'/'diffenrent', and 'subspase'. None of these affect the mathematics, but they detract from the presentation.
[Section 2, Lemma 13] In the proof of Lemma 13, the line 'perpendicular to the lines t1(φ*) and t1(φ*)' should read 't1(φ*) and t2(φ*)'.
[Section 3, Lemma 23] Lemma 23 uses the notation 'C′ := C∩Π' although the hypotheses introduce a set D; the intersection should be with D. The proof is deferred to a separation argument that is not given; if Theorem 22 is completed, this lemma's proof should be included as well.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper is a deductive geometry proof whose central results are derived from first principles.

full rationale

The derivation chain is self-contained. Proposition 2 and Proposition 3 are proved directly from calculus and elementary geometry; Proposition 5 reduces local minimality for a convex set to the strictly convex angular problem via supporting lines, and its proof is given in the text. Proposition 10 and Lemma 13 are likewise proved from Lemma 9, which is proved in full. Theorem 21 reduces the n-dimensional case to the planar case by slicing with supporting hyperplanes, and the transference of local minimality to the slice is argued rather than assumed as an input. Theorems 22 and 25 and Lemma 18 are stated with omitted proofs, but omitted proof is incompleteness, not circularity: none of these statements is used as an input to prove itself. The only self-citation, reference [6] to Kenderov's ICME paper, appears in the appendix's parking example and supports none of the mathematical theorems. There are no fitted constants, no empirical predictions, and no uniqueness theorem imported from the authors' prior work. The advertised longest-chord and exterior-point results depend on omitted proofs, which is a correctness or completeness risk, but not a circularity risk.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The central claim rests on standard convex analysis and the paper's own 2D lemmas. There are no free parameters, fitted constants, or invented entities. The only external load-bearing fact is Rockafellar's theorem on tangents versus unique supporting lines, which is cited and used to pass from smoothness to differentiability.

assumptions (3)

standard math Every boundary point of a compact convex set with nonempty interior has at least one supporting hyperplane; uniqueness of the supporting hyperplane implies differentiability of the boundary at that point.
Used in Sections 2 and 3 to define smoothness and to justify tangent-based proofs (e.g., before Proposition 5 and in Theorem 21).
standard math Rockafellar, Convex Analysis, Theorem 25.1: a convex set has a tangent at a boundary point if and only if there is a unique supporting line there.
Explicitly cited in Section 2 after Lemma 9 to connect the graph-tangent definition with unique supporting lines.
domain assumption The geometric setting: C is a compact convex set with nonempty interior (or a smooth convex set, or a convex polytope), O is either interior to C or outside C with B in [OA] for the exterior case.
This is the problem domain stated in the abstract and Section 1; all theorems are conditional on this setup.

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Cite this review

Pith. "Pith review of A Generalization of a Classical Geometric Extremum Problem." pith.science (2026). https://pith.science/paper/FSBFHLLA

@misc{pith2026250607252,
  author       = {Pith},
  title        = {Pith review of: A Generalization of a Classical Geometric Extremum Problem},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FSBFHLLA}},
  note         = {Machine review of arXiv:2506.07252}
}

abstract

Let $\partial \,\mathcal{C}$ be the boundary of a compact convex body $\mathcal{C}$ in $\mathbb{R}^n,\, n\geq 2$, and $O$ be an interior point of $\mathcal C$. Every straight line $l$ containing $O$ cuts from $\mathcal{C}$ a segment $[AB]$ with end-points on $\partial \,\mathcal{C}$. It is shown that if $[AB]$ is the shortest such segment, then $\partial \,\mathcal{C}$ is smooth at the points $A$ and $ B$ (i.e. at both of them there is only one supporting hyperplane for $\mathcal{C}$) and, something more, the normals to the unique supporting hyperplanes at the points $A$ and $B$ intersect at a point belonging to the hiperplane through $O$ which is orthogonal to $[AB]$. If $\mathcal{C}$ is a smooth compact convex body in $\mathbb{R}^n,\, n\geq 2$, the above property holds also when $[AB]$ is the longest such segment. Similar results have place also when $O$ is outside the set $\mathcal{C}$. The ``local versions'' of these results (when the length $|AB|$ of the segment $[AB]$ is locally maximal or locally minimal) also have a place. More specific results are obtained in the particular case when $\mathcal{C}$ is a convex polytope.

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Reference graph

Works this paper leans on

14 extracted references · 13 canonical work pages

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [6]

P. S. Kenderov Powering Knowledge Versus Pouring Facts , In G. Kaiser et. al. (eds.). In- vited Lectures from the 13th International Congress on Mathematical Education p. 289 – 306, (https://www.springer.com/gp/book/9783319721699)

arXiv

[2] [1]

Bottema, Versheidenheden, Euclides, Maanblad voor de Didaktiek van de Wiskunde, Sep- tember (1968) p

O. Bottema, Versheidenheden, Euclides, Maanblad voor de Didaktiek van de Wiskunde, Sep- tember (1968) p. 20-25

work page 1968

[3] [2]

A Sequel to the First Six Books of the Elements of Euclid Hodges,Figis, and Co, Dublin, 1886

Casey, J. A Sequel to the First Six Books of the Elements of Euclid Hodges,Figis, and Co, Dublin, 1886

work page

[4] [3]

Introduction to Geometry, Wiley, New York, 1961 (lst ed.),

Coxeter, H.S.M. Introduction to Geometry, Wiley, New York, 1961 (lst ed.),

work page 1961

[5] [4]

Coxeter , Jan van de Craats, Philon Lines in Non-Eucledean Planes , Journal of Geom- etry , Vol

H.S.M. Coxeter , Jan van de Craats, Philon Lines in Non-Eucledean Planes , Journal of Geom- etry , Vol. 48 (1993) , p. 26 - 55

work page 1993

[6] [5]

Eves, Philo’s line , Scripta Mathematica 24, 1959, p

H. Eves, Philo’s line , Scripta Mathematica 24, 1959, p. 141-148

work page 1959

[7] [7]

Lhuilier Principiorum calculi differentialis et integralis expositio elementaris, Tubingae,1795, p

M. Lhuilier Principiorum calculi differentialis et integralis expositio elementaris, Tubingae,1795, p. 270

work page

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Lhuilier Solution d’un probl` eme de g´ eom´ etrie, d´ ependant de la th´ eorie des maximis et min- imis, Annales de Math´ ematiques pures et appliqu´ ees, tome 2 (1811-1812), p

M. Lhuilier Solution d’un probl` eme de g´ eom´ etrie, d´ ependant de la th´ eorie des maximis et min- imis, Annales de Math´ ematiques pures et appliqu´ ees, tome 2 (1811-1812), p. 17-22

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