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arxiv: 1907.01868 · v1 · submitted 2019-07-03 · 🧮 math.MG · math.PR

On random approximations by generalized disc-polygons

Ferenc Fodor , D\'aniel I. Papv\'ari , Viktor V\'igh This is my paper

Pith reviewed 2026-05-25 09:51 UTC · model grok-4.3

classification 🧮 math.MG math.PR
keywords L-convexityrandom approximationconvex discsexpected verticesmissed areageneralized convexityconstant width
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The pith

When the convex disc equals its own shape L, the expected number of vertices in its random L-convex approximation approaches a finite limit determined only by L.

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

The paper defines L-convexity for a convex disc K relative to another disc L, where K is the intersection of all L-translates containing it. It then samples n random points uniformly in K and forms the smallest L-convex set containing them, called K_{(n)}. For the case K equal to L, the expected vertex count of K_{(n)} converges to a finite number as n grows, unlike the logarithmic growth in standard convexity. This holds under C^2 smoothness with positive curvature, and the paper derives formulas that extend the circular case. A reader might care because it reveals how the choice of L can cap the complexity of random polygonal approximations inside itself.

Core claim

When K equals L, both C^2_+ smooth convex discs with K L-convex, the expected number of vertices f_0(K_{(n)}) of the random L-convex polygon tends to a finite limit depending only on L as n tends to infinity. The paper also determines the maximum and minimum of this limit over all convex discs L of constant width 1. These results generalize the r-spindle convex case where L is a circle.

What carries the argument

The random L-convex polygon K_{(n)}, obtained as the intersection of all translates of L that contain the n i.i.d. uniform random points from K.

If this is right

  • When K = L the expected vertex number converges to a finite limit set by L.
  • The missed area expectation behaves similarly to classical cases under curvature conditions.
  • Explicit formulas for the limits generalize those for spindle-convex sets.
  • For constant width 1 discs the limit attains specific extrema depending on the shape of L.

Where Pith is reading between the lines

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

  • The stabilization of vertex count suggests that L-convexity creates a rigid global structure preventing indefinite refinement by random points.
  • Similar finite limits might appear in other generalized convexity notions where the approximating sets are constrained by a fixed shape L.
  • One could compute the limit numerically for specific non-circular L to verify the dependence only on L.

Load-bearing premise

K and L are twice continuously differentiable convex discs with positive curvature and K is L-convex.

What would settle it

For a specific smooth convex disc L of constant width, numerically simulate many realizations of L_{(n)} for very large n and check whether the average number of vertices stabilizes around a constant rather than continuing to increase.

read the original abstract

For two convex discs $K$ and $L$, we say that $K$ is $L$-convex if it is equal to the intersection of all translates of $L$ that contain $K$. In $L$-convexity the set $L$ plays a similar role as closed half-spaces do in the classical notion of convexity. We study the following probability model: Let $K$ and $L$ be $C^2_+$ smooth convex discs such that $K$ is $L$-convex. Select $n$ i.i.d. uniform random points $x_1,\ldots, x_n$ from $K$, and consider the intersection $K_{(n)}$ of all translates of $L$ that contain all of $x_1,\ldots, x_n$. The set $K_{(n)}$ is a random $L$-convex polygon in $K$. We study the expectation of the number of vertices $f_0(K_{(n)})$ and the missed area $A(K\setminus K_{n})$ as $n$ tends to infinity. We consider two special cases of the model. In the first case we assume that the maximum of the curvature of the boundary of $L$ is strictly less than $1$ and the minimum of the curvature of $K$ is larger than $1$. In this setting the expected number of vertices and missed area behave in a similar way as in the classical convex case and in the $r$-spindle convex case (when $L$ is a radius $r$ circular disc). The other case we study is when $K=L$. This setting is special in the sense that an interesting phenomenon occurs: the expected number of vertices tends to a finite limit depending only on $L$. This was previously observed in the special case when $L$ is a circle of radius $r$ (Fodor, Kevei and V\'igh (2014)). We also determine the extrema of the limit of the expectation of the number of vertices of $L_{(n)}$ if $L$ is a convex discs of constant width $1$. The formulas we prove can be considered as generalizations of the corresponding $r$-spindle convex statements proved by Fodor, Kevei and V\'igh (2014).

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 paper defines L-convexity for convex discs K and L, where K is the intersection of all L-translates containing it. It considers n i.i.d. uniform random points in K and the random L-convex polygon K_{(n)} as the intersection of all L-translates containing those points. The main results concern the asymptotics of E[f_0(K_{(n)})] and the missed area A(K K_{(n)}) as n→∞ under C^2_+ smoothness. Two regimes are treated: (i) max curvature of L <1 and min curvature of K >1, where the quantities behave as in the classical convex and r-spindle cases; (ii) the case K=L, where E[f_0(K_{(n)})] converges to a finite limit depending only on L (generalizing the circular case of Fodor-Kevei-Vigh 2014). Extrema of this limit are computed when L has constant width 1.

Significance. If the derivations hold, the work supplies new limit theorems for random approximation in the setting of L-convexity, a natural generalization of ordinary convexity and spindle convexity. The finite-limit phenomenon when K=L is the most distinctive contribution; the explicit extrema for constant-width bodies and the reduction to the known circular case are concrete additions. The C^2_+ hypothesis supplies the uniform quadratic estimates needed for summability of vertex probabilities.

minor comments (3)
  1. The abstract states that the formulas 'generalize the r-spindle convex statements,' but the introduction should explicitly recall the precise statements from Fodor-Kevei-Vigh (2014) that are being extended (e.g., the explicit constant for the circular limit).
  2. Notation: the random set is written both as K_{(n)} and K_n in the abstract; a single consistent symbol should be used throughout.
  3. The curvature conditions (max κ_L <1, min κ_K >1) are stated only in the abstract; they should be restated verbatim at the beginning of the section that treats the first regime.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No major comments were provided in the report, so we have no specific points to address point-by-point. We will make any minor editorial or typographical adjustments as needed in the revised version.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper derives new asymptotic formulas for E[f_0(K_{(n)})] and missed area under C^2_+ smoothness and L-convexity assumptions. The central limit result when K=L is proved directly from cap-area estimates that rely on curvature bounds, not on any fitted quantity or prior result. The 2014 citation (overlapping authors) is invoked only to note the special-case observation being generalized; the present derivations are self-contained and do not reduce to that citation by construction. No self-definitional, fitted-input, or ansatz-smuggling steps appear.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no explicit free parameters, axioms, or invented entities; all technical assumptions are smoothness and curvature inequalities on K and L.

pith-pipeline@v0.9.0 · 5964 in / 1008 out tokens · 42802 ms · 2026-05-25T09:51:20.710391+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

30 extracted references · 30 canonical work pages

  1. [1]

    B´ ar´ any,Random points and lattice points in convex bodies , Bull

    I. B´ ar´ any,Random points and lattice points in convex bodies , Bull. Amer. Math. Soc. (N.S.) 45 (2008), no. 3, 339–365

    work page 2008

  2. [2]

    Bezdek, Z

    K. Bezdek, Z. L´ angi, M. Nasz´ odi, and P. Papez, Ball-polyhedra, Discrete Comput. Geom. 38 (2007), no. 2, 201–230

    work page 2007

  3. [3]

    Blaschke, Kreis und Kugel , W alter de Gruyter & Co., Berlin, 1956 (German)

    W. Blaschke, Kreis und Kugel , W alter de Gruyter & Co., Berlin, 1956 (German). 2te Aufl

    work page 1956

  4. [4]

    K. J. B¨ or¨ oczky, F. Fodor, M. Reitzner, and V. V ´ ıgh,Mean width of random polytopes in a reasonably smooth convex body , J. Multivariate Anal. 100 (2009), no. 10, 2287–2295

    work page 2009

  5. [5]

    Efron, The convex hull of a random set of points , Biometrika 52 (1965), 331–343

    B. Efron, The convex hull of a random set of points , Biometrika 52 (1965), 331–343

    work page 1965

  6. [6]

    Fejes T´ oth, Packing of r-convex discs , Studia Sci

    L. Fejes T´ oth, Packing of r-convex discs , Studia Sci. Math. Hungar. 17 (1982), no. 1-4, 449–452

    work page 1982

  7. [7]

    Fejes T´ oth, Packing and covering with r-convex discs , Studia Sci

    L. Fejes T´ oth, Packing and covering with r-convex discs , Studia Sci. Math. Hungar. 18 (1982), no. 1, 69–73

    work page 1982

  8. [8]

    Fejes T´ oth and F

    G. Fejes T´ oth and F. Fodor, Dowker-type theorems for hyperconvex discs , Period. Math. Hungar. 70 (2015), no. 2, 131–144

    work page 2015

  9. [9]

    Fodor, Random ball-polytopes in smooth convex bodies , arXiv:1906.11480 (2019)

    F. Fodor, Random ball-polytopes in smooth convex bodies , arXiv:1906.11480 (2019)

    work page arXiv 1906

  10. [10]

    Fodor, P

    F. Fodor, P. Kevei, and V. V ´ ıgh, On random disc polygons in smooth convex discs , Adv. in Appl. Probab. 46 (2014), no. 4, 899–918

    work page 2014

  11. [11]

    Fodor, ´A

    F. Fodor, ´A. Kurusa, and V. V ´ ıgh,Inequalities for hyperconvex sets, Adv. in Geom. 16 (2016), no. 3, 337–348

    work page 2016

  12. [12]

    Fodor and V

    F. Fodor and V. V ´ ıgh, Disc-polygonal approximations of planar spindle convex se ts, Acta Sci. Math. (Szeged) 78 (2012), no. 1-2, 331–350

    work page 2012

  13. [13]

    Fodor and V

    F. Fodor and V. V ´ ıgh, Variance estimates for random disc-polygons in smooth conv ex discs , J. Appl. Probab. 55 (2018), no. 4, 1143–1157

    work page 2018

  14. [14]

    Hug, Measures, curvatures and currents in convex geometry , Habilitationsschrift, Albert Ludwigs Universit¨ at Freiburg, 1999

    D. Hug, Measures, curvatures and currents in convex geometry , Habilitationsschrift, Albert Ludwigs Universit¨ at Freiburg, 1999

    work page 1999

  15. [15]

    Hug, Random polytopes, Stochastic geometry, spatial statistics and random fields , Lecture Notes in Math., vol

    D. Hug, Random polytopes, Stochastic geometry, spatial statistics and random fields , Lecture Notes in Math., vol. 2068, Springer, Heidelberg, 2013, pp. 2 05–238

    work page 2068

  16. [16]

    L´ angi, M

    Z. L´ angi, M. Nasz´ odi, and I. Talata,Ball and spindle convexity with respect to a convex body , Aequationes Math. 85 (2013), no. 1-2, 41–67

    work page 2013

  17. [17]

    Martini, L

    H. Martini, L. Montejano, and D. Oliveros, Bodies of constant width , Birkh¨ auser/Springer, Cham, 2019. An introduction to convex geometry with applica tions

    work page 2019

  18. [18]

    A. E. Mayer, Eine ¨Uberkonvexit¨ at, Math. Z. 39 (1935), 511–531

    work page 1935

  19. [19]

    Papv´ ari,Konvex lemezek v´ eletlen eltoltjainak metszete (On intersections of random trans- lates of a convex disc) , 2019 (Hungarian)

    D. Papv´ ari,Konvex lemezek v´ eletlen eltoltjainak metszete (On intersections of random trans- lates of a convex disc) , 2019 (Hungarian). Thesis (Bachelor’s)–University of Sze ged, Hungary

    work page 2019

  20. [20]

    Paouris and P

    G. Paouris and P. Pivovarov, Randomized isoperimetric inequalities , Convexity and concen- tration, IMA Vol. Math. Appl., vol. 161, Springer, New York, 2017, pp. 391–425

    work page 2017

  21. [21]

    Reitzner, Random polytopes, New perspectives in stochastic geometry, Oxford Univ

    M. Reitzner, Random polytopes, New perspectives in stochastic geometry, Oxford Univ. Pre ss, Oxford, 2010, pp. 45–76

    work page 2010

  22. [22]

    R´ enyi and R

    A. R´ enyi and R. Sulanke, ¨Uber die konvexe H¨ ulle von n zuf¨ allig gew¨ ahlten Punkten, Z. W ahrscheinlichkeitstheorie und Verw. Gebiete 2 (1963), 75–84 (1963) (German)

    work page 1963

  23. [23]

    R´ enyi and R

    A. R´ enyi and R. Sulanke, ¨Uber die konvexe H¨ ulle von n zuf¨ allig gew¨ ahlten Punkten. II, Z. W ahrscheinlichkeitstheorie und Verw. Gebiete 3 (1964), 138–147 (1964) (German)

    work page 1964

  24. [24]

    R´ enyi and R

    A. R´ enyi and R. Sulanke, Zuf¨ allige konvexe Polygone in einem Ringgebiet , Z. W ahrschein- lichkeitstheorie und Verw. Gebiete 9 (1968), 146–157 (German)

    work page 1968

  25. [25]

    L. A. Santal´ o, On plane hyperconvex figures , Summa Brasil. Math. 1 (1946), 221–239 (1948) (Spanish, with English summary)

    work page 1946

  26. [26]

    Schneider, Convex bodies: the Brunn-Minkowski theory , Second expanded edition, Encyclo- pedia of Mathematics and its Applications, vol

    R. Schneider, Convex bodies: the Brunn-Minkowski theory , Second expanded edition, Encyclo- pedia of Mathematics and its Applications, vol. 151, Cambri dge University Press, Cambridge, 2014

    work page 2014

  27. [27]

    Schneider, Discrete aspects of stochastic geometry , Handbook of discrete and computa- tional geometry, 3rd ed., CRC, Boca Raton, FL, 2018, pp

    R. Schneider, Discrete aspects of stochastic geometry , Handbook of discrete and computa- tional geometry, 3rd ed., CRC, Boca Raton, FL, 2018, pp. 299– 329

    work page 2018

  28. [28]

    Schneider, Recent results on random polytopes , Boll

    R. Schneider, Recent results on random polytopes , Boll. Unione Mat. Ital. (9) 1 (2008), no. 1, 17–39

    work page 2008

  29. [29]

    Schneider and W

    R. Schneider and W. W eil, Stochastic and integral geometry , Probability and its Applications (New York), Springer-Verlag, Berlin, 2008. 16 FERENC FODOR 1, D ´ANIEL I. PAPV ´ARI2, AND VIKTOR V ´IGH3

    work page 2008

  30. [30]

    W eil and J

    W. W eil and J. A. Wieacker, Stochastic geometry, Handbook of convex geometry, Vol. A, B, North-Holland, Amsterdam, 1993, pp. 1391–1438. Department of Geometry, Bolyai Institute, University of Sze ged, Aradi v ´ertan´uk tere 1, 6720 Szeged, Hungary E-mail address : fodorf@math.u-szeged.hu University of Szeged, Aradi v ´ertan´uk tere 1, 6720 Szeged, Hungar...

    work page 1993