On a special class of equidistant sets in the Euclidean space

\'A. Nagy; Cs. Vincze; M. Ol\'ah; M. Stoika

arxiv: 2605.22451 · v1 · pith:KEX57JCInew · submitted 2026-05-21 · 🧮 math.MG

On a special class of equidistant sets in the Euclidean space

\'A. Nagy , M. Ol\'ah , M. Stoika , Cs. Vincze This is my paper

Pith reviewed 2026-05-22 01:46 UTC · model grok-4.3

classification 🧮 math.MG

keywords equidistant setsequidistant functionsfocal setsEuclidean spacegraphs of functionsspherical approximationminimum operator

0 comments

The pith

Replacing a hyperplane with a sphere as focal set produces a new class of equidistant functions.

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

The paper generalizes earlier work on equidistant sets by letting one focal set be approximated by a sphere instead of a flat hyperplane. Points at equal distance from the sphere and from the epigraph of a second function then form the graph of an equidistant function over the sphere. The authors give necessary and sufficient conditions for existence along vertical lines, introduce upper and lower versions of these functions, and show that the construction commutes with the minimum operator.

Core claim

Substituting the horizontal hyperplane by a sphere as the second-order approximation for one focal set yields a new type of equidistant function whose graph consists of points equidistant to both focal sets; the paper establishes general existence criteria, upper and lower variants, and a commuting property with the minimum operator.

What carries the argument

Equidistant function defined as the graph over the sphere of points having equal distances to the sphere and to the epigraph of a positive continuous function.

If this is right

Vertical lines through the sphere intersect the equidistant set at most twice, permitting well-defined upper and lower equidistant functions.
Existence of an equidistant function is equivalent to a necessary and sufficient condition stated in terms of the two focal sets.
The equidistant-function operator commutes with the pointwise minimum of the defining functions.

Where Pith is reading between the lines

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

The spherical version may give more accurate local models when one focal set has noticeable curvature.
The same replacement idea could be iterated to higher-order quadratic or cubic approximations for still more curved focal sets.

Load-bearing premise

The equidistant points form the graph of a function over the sphere or its hyperplane replacement.

What would settle it

A concrete pair consisting of a sphere and the epigraph of a continuous positive function whose equal-distance set fails to be the graph of any single-valued function over the sphere would show the generalization does not hold in general.

Figures

Figures reproduced from arXiv: 2605.22451 by \'A. Nagy, Cs. Vincze, M. Ol\'ah, M. Stoika.

**Figure 4.** Figure 4: Some examples demonstrating the cases when (A) there are no critical parameters, so the parameter domain is D = ]−∞, ∞[; (B) there are only positive critical parameters: D = ]−∞, t+ 0 [; (C) there are both negative and positive critical parameters: D = ]t − 0 , t+ 0 [. 3.1.2. The expression of the functions f and f ′ in terms of the equidistant parameterization (4) and (5). According to formulas (7) and (8… view at source ↗

**Figure 5.** Figure 5: The signed distance functions α and αu in case of f(t1, t2) = t 2 1 + 20t1t2 + 1000t 2 2 − 30t2 + 1 with u = (1, 0). Therefore y(t) = f(t) − (r(t) − R) p 1 − |g(t)| 2 and we have (29) f(t) = y(t) + (r(t) − R) p 1 − |g(t)| 2 . On the other hand (30) ∇f(t) = g(t) p 1 − |g(t)| 2 . 3.2.3. The case of a twice continuously differentiable function: the compatibility of the equidistant parameterization. Let f be a… view at source ↗

read the original abstract

An equidistant set in the Euclidean space consists of points having equal distances to both members of a given pair of sets, called focal sets. Since there is no effective formula to compute the distance of a point and a set, it is hard to determine the points of an equidistant set in general. Therefore, it is important to investigate some special cases. In the paper we investigate equidistant sets that can be given as the graph of a function. They are called equidistant functions. In the previously examined conceptual model, one of the focal sets is the horizontal hyperplane through the origin and the other one is the epigraph of a positive-valued, continuous function. The equidistant points form the graph of another function over the hyperplane. In a general situation, the hyperplane is the first-order (linear) approximation for one of the focal sets. A natural idea is to substitute the hyperplane by a circle (sphere) as a second-order (quadratic) approximation for one of the focal sets in more complicated cases. Such a generalization results in a new type of equidistant functions we are going to investigate in the present paper. Before considering the special cases in detail, we present some general observations: a necessary and sufficient condition for the existence of equidistant points along the vertical lines, upper/lower equidistant functions, equidistant functions, a necessary and sufficient condition for the existence of the equidistant function, equidistant functions and the minimum operator (a kind of commuting property).

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.

Desk Editor's Note private letter to a colleague

The paper extends equidistant functions to spherical focal sets as a quadratic approximation but the abstract provides conditions without derivations, leaving the adaptation to nonlinearity unverified in the visible text.

read the letter

Dear colleague, The main point to know is that the authors generalize their equidistant functions by using a sphere instead of a hyperplane as one focal set, presenting it as a second-order approximation that yields a new type of these functions. They do a good job setting up the general observations first. These include necessary and sufficient conditions for equidistant points along vertical lines, the notions of upper and lower equidistant functions, a condition for the existence of the full equidistant function, and how it interacts with the minimum operator through a commuting property. This keeps the work consistent with their previous hyperplane-based model. The spherical case is framed as a natural next step for more complicated situations where a linear approximation is not enough. That extension is the actual novelty here. The soft spots are mostly about verification. The abstract lists the conditions but does not provide the derivations or explicit formulas, making it difficult to confirm soundness right away. The stress-test concern about vertical lines potentially having more than one intersection due to the nonlinear distance to the sphere is worth paying attention to. The paper applies the same vertical restriction and general condition without showing a re-derivation for the curved focal set. If the full manuscript handles this by proving uniqueness still holds or by adjusting the conditions, that would strengthen it. Otherwise it could be a real issue. No obvious circularity or invented entities appear in the description. This paper is for a small group of specialists in metric geometry who study equidistant sets and their representations as graphs. A reader already familiar with the hyperplane version might get some value from seeing how the ideas carry over, but it probably won't interest a wider audience. I recommend sending it for peer review. The structure is there and the generalization is logical, so referees can check the details and suggest improvements where needed. Best regards,

Referee Report

2 major / 2 minor

Summary. The paper investigates equidistant sets in Euclidean space that can be expressed as the graph of a function (equidistant functions). It generalizes an earlier model with a horizontal hyperplane as one focal set by replacing the hyperplane with a sphere as a second-order approximation, and states general observations on necessary-and-sufficient conditions for existence of equidistant points along vertical lines, upper/lower equidistant functions, a condition for the full equidistant function, and a commuting property with the minimum operator.

Significance. If the central claims are established with explicit derivations, the work would introduce a new class of equidistant functions adapted to quadratic approximations of focal sets. This could be useful for studying geometric loci in approximation theory and for constructing explicit examples where distance functions are nonlinear.

major comments (2)

[General observations] General observations (prior to special cases): the necessary-and-sufficient condition for existence of equidistant points along vertical lines is stated for the linear (hyperplane) focal set. When the focal set is replaced by a sphere the distance is nonlinear, so the equidistance equation along a fixed-x vertical line need not have at most one root; the manuscript does not re-derive or verify the condition under this curvature.
[Special cases] Definition of equidistant functions via vertical lines: the construction assumes that restricting to vertical lines yields a well-defined graph (upper/lower functions). For a spherical focal set this uniqueness is not automatic and is not shown to survive the generalization.

minor comments (2)

[General observations] The phrase 'a kind of commuting property' for the minimum operator should be replaced by a precise statement of the claimed identity.
[Introduction] Notation for the spherical focal set (center, radius) should be introduced explicitly before the special-case sections.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address the major comments point by point below, agreeing that explicit verification is needed for the spherical case, and outline the planned revisions.

read point-by-point responses

Referee: [General observations] General observations (prior to special cases): the necessary-and-sufficient condition for existence of equidistant points along vertical lines is stated for the linear (hyperplane) focal set. When the focal set is replaced by a sphere the distance is nonlinear, so the equidistance equation along a fixed-x vertical line need not have at most one root; the manuscript does not re-derive or verify the condition under this curvature.

Authors: We appreciate this observation. The general observations are presented prior to the special cases and are intended to apply in the generalized spherical setting. However, we acknowledge that the derivation of the necessary and sufficient condition for equidistant points along vertical lines was developed in the linear case, and the nonlinearity of the distance to the sphere requires explicit re-derivation to confirm it still yields at most one root per line. We will revise the manuscript to include this verification. revision: yes
Referee: [Special cases] Definition of equidistant functions via vertical lines: the construction assumes that restricting to vertical lines yields a well-defined graph (upper/lower functions). For a spherical focal set this uniqueness is not automatic and is not shown to survive the generalization.

Authors: We agree that uniqueness along vertical lines is not automatic with a curved focal set and must be established for the graph to be well-defined. In the special cases section we proceed from the stated continuity and positivity assumptions on the functions, but we will add an explicit argument or lemma showing that these conditions suffice to guarantee at most one solution per vertical line even when one focal set is spherical. revision: yes

Circularity Check

0 steps flagged

No circularity: general observations stated independently of special cases

full rationale

The paper states a necessary and sufficient condition for equidistant points along vertical lines as a general observation before specializing to the spherical focal set. This condition is presented as an independent geometric criterion rather than derived from or equivalent to the hyperplane case by construction. No equations reduce a claimed result to a fitted input, self-citation chain, or renamed definition. The generalization to spheres is motivated separately as a quadratic approximation, with the vertical-line graph property invoked only where the existence condition holds. The derivation chain remains self-contained without load-bearing self-references or definitional loops.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The review rests on the abstract alone; no explicit free parameters, invented entities, or additional axioms beyond the geometric approximation assumption are visible.

axioms (1)

domain assumption Focal sets admit first-order (hyperplane) and second-order (sphere) approximations that allow the equidistant set to be expressed as a graph of a function.
Invoked when the authors substitute the hyperplane by a circle or sphere and restrict to vertical lines.

pith-pipeline@v0.9.0 · 5811 in / 1151 out tokens · 47014 ms · 2026-05-22T01:46:58.298879+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking echoes

?

echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

A natural idea is to substitute the hyperplane by a circle (sphere) as a second-order (quadratic) approximation... equidistant functions... Theorems 5-8, Theorems 10-11... Theorem 9, Theorem 12
IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean absolute_floor_iff_bare_distinguishability unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

the vertical line through x contains an equidistant point... necessary and sufficient condition... G+(x), G-(x)... equidistant function

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages

[1]

P.\ Erd o s and I.\ Vincze, On the approximation of closed convex plane curves, Mat.\ Lapok 9 (1958), 19-36 (in Hungarian, summaries in Russian and German)

work page 1958
[2]

Gross and T.-K

C. Gross and T.-K. Strempel, On generalizations of conics and on a generalization of the Fermat-Toricelli problem, Amer. Math. Monthly, 105 (1998), no. 8, 732-743

work page 1998
[3]

S.\ R.\ Lay, Convex Sets and Their Applications, John Wiley & Sons, Inc., 1982

work page 1982
[4]

L. D. Loveland, When midsets are manifolds, Proc. Amer. Math. Soc. 61 (2), 1976, 353-360

work page 1976
[5]

Polyellipses and optimization, Quart

Z.\ A.\ Melzak and J.\ S.\ Forsyth, Polyconics 1. Polyellipses and optimization, Quart. Appl. Math. 35 , Vol. 2 (1977), 239-255

work page 1977
[6]

Ponce, P

M. Ponce, P. Santib\'a\ nez, On equidistant sets and generalized conics: the old and the new, Amer. Math. Monthly, 121 (1) 2014, pp. 18-32

work page 2014
[7]

\ Nagy, Examples and notes on generalized conics and their applications, AMAPN Vol.\ 26 (2010), 359-575

Cs.\ Vincze and \' A . \ Nagy, Examples and notes on generalized conics and their applications, AMAPN Vol.\ 26 (2010), 359-575

work page 2010
[8]

Vol.\ 61 (2011), 815-828

Cs.\ Vincze and \' A .\ Nagy, An introduction to the theory of generalized conics and their applications, Journal of Geom.\ and Phys. Vol.\ 61 (2011), 815-828

work page 2011
[9]

of Approx

Cs.\ Vincze and \' A .\ Nagy, On the theory of generalized conics with applications in geometric tomography, J. of Approx. Theory 164 (2012), 371-390

work page 2012
[10]

Ol\' a h, \ L.\ F\' o ri\' a n and S.\ L o rinc, On computable classes of equidistant sets: finite focal sets, Involve - a Journal of Math., Vol

Cs.\ Vincze, \ A.\ Varga,\ M. Ol\' a h, \ L.\ F\' o ri\' a n and S.\ L o rinc, On computable classes of equidistant sets: finite focal sets, Involve - a Journal of Math., Vol. 11 (2018), No. 2, pp. 271–282

work page 2018
[11]

Vincze, A

Cs. Vincze, A. Varga, M. Ol\' a h, L. F\' o ri\' a n, On computable classes of equidistant sets: equidistant functions, Miskolc Math. Notes, Vol. 19 (2018), No. 1, pp. 677-689

work page 2018
[12]

Vincze, M

Cs. Vincze, M. Oláh, L. Lengyel, On equidistant polytopes in the Euclidean space, Involve - a Journal of Math., Vol. 13 (2020), No. 4, pp. 577–595, DOI: 10.2140/involve.2020.13.577

work page doi:10.2140/involve.2020.13.577 2020
[13]

\' A . Nagy, M. Ol\' a h, M. Stoika, Cs. Vincze, On computable classes of equidistant sets: multivariate equidistant functions, Aequat. Math. 100, 22 (2026). https://doi.org/10.1007/s00010-025-01260-8, arXiv:2503.05901

work page doi:10.1007/s00010-025-01260-8 2026
[14]

J. B. Wilker, Equidistant sets and their connectivity properties, Proc. Amer. Math. Soc. 47 (2), 1975, 446-452

work page 1975

[1] [1]

P.\ Erd o s and I.\ Vincze, On the approximation of closed convex plane curves, Mat.\ Lapok 9 (1958), 19-36 (in Hungarian, summaries in Russian and German)

work page 1958

[2] [2]

Gross and T.-K

C. Gross and T.-K. Strempel, On generalizations of conics and on a generalization of the Fermat-Toricelli problem, Amer. Math. Monthly, 105 (1998), no. 8, 732-743

work page 1998

[3] [3]

S.\ R.\ Lay, Convex Sets and Their Applications, John Wiley & Sons, Inc., 1982

work page 1982

[4] [4]

L. D. Loveland, When midsets are manifolds, Proc. Amer. Math. Soc. 61 (2), 1976, 353-360

work page 1976

[5] [5]

Polyellipses and optimization, Quart

Z.\ A.\ Melzak and J.\ S.\ Forsyth, Polyconics 1. Polyellipses and optimization, Quart. Appl. Math. 35 , Vol. 2 (1977), 239-255

work page 1977

[6] [6]

Ponce, P

M. Ponce, P. Santib\'a\ nez, On equidistant sets and generalized conics: the old and the new, Amer. Math. Monthly, 121 (1) 2014, pp. 18-32

work page 2014

[7] [7]

\ Nagy, Examples and notes on generalized conics and their applications, AMAPN Vol.\ 26 (2010), 359-575

Cs.\ Vincze and \' A . \ Nagy, Examples and notes on generalized conics and their applications, AMAPN Vol.\ 26 (2010), 359-575

work page 2010

[8] [8]

Vol.\ 61 (2011), 815-828

Cs.\ Vincze and \' A .\ Nagy, An introduction to the theory of generalized conics and their applications, Journal of Geom.\ and Phys. Vol.\ 61 (2011), 815-828

work page 2011

[9] [9]

of Approx

Cs.\ Vincze and \' A .\ Nagy, On the theory of generalized conics with applications in geometric tomography, J. of Approx. Theory 164 (2012), 371-390

work page 2012

[10] [10]

Ol\' a h, \ L.\ F\' o ri\' a n and S.\ L o rinc, On computable classes of equidistant sets: finite focal sets, Involve - a Journal of Math., Vol

Cs.\ Vincze, \ A.\ Varga,\ M. Ol\' a h, \ L.\ F\' o ri\' a n and S.\ L o rinc, On computable classes of equidistant sets: finite focal sets, Involve - a Journal of Math., Vol. 11 (2018), No. 2, pp. 271–282

work page 2018

[11] [11]

Vincze, A

Cs. Vincze, A. Varga, M. Ol\' a h, L. F\' o ri\' a n, On computable classes of equidistant sets: equidistant functions, Miskolc Math. Notes, Vol. 19 (2018), No. 1, pp. 677-689

work page 2018

[12] [12]

Vincze, M

Cs. Vincze, M. Oláh, L. Lengyel, On equidistant polytopes in the Euclidean space, Involve - a Journal of Math., Vol. 13 (2020), No. 4, pp. 577–595, DOI: 10.2140/involve.2020.13.577

work page doi:10.2140/involve.2020.13.577 2020

[13] [13]

\' A . Nagy, M. Ol\' a h, M. Stoika, Cs. Vincze, On computable classes of equidistant sets: multivariate equidistant functions, Aequat. Math. 100, 22 (2026). https://doi.org/10.1007/s00010-025-01260-8, arXiv:2503.05901

work page doi:10.1007/s00010-025-01260-8 2026

[14] [14]

J. B. Wilker, Equidistant sets and their connectivity properties, Proc. Amer. Math. Soc. 47 (2), 1975, 446-452

work page 1975