On Random Disc Polygons in Smooth Convex Discs

Fodor, Ferenc; Kevei, Péter; Vígh, Viktor

doi:10.1017/s0001867800007473

Cited by 9 publications

(57 citation statements)

References 20 publications

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“…Reuleaux‐polygons are not smooth, however with a slight modification at the vertices one can construct a smooth convex disc of constant width 1 such that the limit of the expectation of the number of the vertices is arbitrarily close to 2π 2 . The lower bound is achieved when L is a circle, as it was shown in [10, Theorem 1.3].…”

Section: Introduction and Resultsmentioning

confidence: 67%

“…Recently, another probability model of random polytopes emerged that is based on intersections of congruent closed balls of suitable radius, see [9, 10, 13]. For a fixed

r > 0

, a convex disc

K \subset {double-struckR}^{2}

is called r ‐spindle convex (sometimes also called r ‐hyperconvex [8] or r ‐convex [6, 7]) if, together with any two points

x, y \in K

, the set

{false[x, y false]}_{r}

, consisting of all shorter circular arcs of radius at least r and connecting x and y , is contained in K .…”

Section: Introduction and Resultsmentioning

confidence: 99%

“…In a recent paper, Fodor et al . [10] proved asymptotic formulas for the expectation of the number of vertices, missed area, and perimeter difference of

K_{false(n false)}^{r}

under suitable smoothness assumption on the boundary of K . These asymptotic formulas are generalizations of the corresponding classical results of Rényi and Sulanke in the limit as

r \to \infty

.…”

Section: Introduction and Resultsmentioning

confidence: 99%

“…Note that in both Theorem 1.1 and Corollary 1.1, we get back the corresponding statements of Fodor et al . [10, Theorem 1.1] when L is a circle of radius

r > 1

.…”

Section: Introduction and Resultsmentioning

confidence: 99%

“…Namely, the expectation of the number of vertices tends to a finite limit determined by only L . This has already been pointed out in the case when

L = B^{2}

in the paper by Fodor et al ., see [10, Theorem 1.3], and in d dimensions for

L = B^{d}

by Fodor [9, Theorem 1.1].…”

Section: Introduction and Resultsmentioning

confidence: 99%

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On Random Approximations by Generalized Disc‐polygons

2020

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For two convex discs K and L, we say that K is L-convex (Lángi et al., Aequationes Math. 85(1-2) (2013), 41-67) 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 independent and identically distributed uniform random points x 1 , . . . , x n from K, and consider the intersection K (n) of all translates of L that contain all of x 1 , . . . , 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 \ 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), see (Fodor et al., Adv. in Appl. Probab. 46 (4) (2014), [899][900][901][902][903][904][905][906][907][908][909][910][911][912][913][914][915][916][917][918]. 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 in Fodor et al. (Adv. in Appl. Probab. 46 (4) (2014), 899-918). 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 et al. in (Adv. in Appl. Probab. 46(4) (2014), 899-918).

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Section: Introduction and Resultsmentioning

confidence: 67%

“…Recently, another probability model of random polytopes emerged that is based on intersections of congruent closed balls of suitable radius, see [9, 10, 13]. For a fixed

r > 0

, a convex disc

K \subset {double-struckR}^{2}

is called r ‐spindle convex (sometimes also called r ‐hyperconvex [8] or r ‐convex [6, 7]) if, together with any two points

x, y \in K

, the set

{false[x, y false]}_{r}

, consisting of all shorter circular arcs of radius at least r and connecting x and y , is contained in K .…”

Section: Introduction and Resultsmentioning

confidence: 99%

“…In a recent paper, Fodor et al . [10] proved asymptotic formulas for the expectation of the number of vertices, missed area, and perimeter difference of

K_{false(n false)}^{r}

under suitable smoothness assumption on the boundary of K . These asymptotic formulas are generalizations of the corresponding classical results of Rényi and Sulanke in the limit as

r \to \infty

.…”

Section: Introduction and Resultsmentioning

confidence: 99%

“…Note that in both Theorem 1.1 and Corollary 1.1, we get back the corresponding statements of Fodor et al . [10, Theorem 1.1] when L is a circle of radius

r > 1

.…”

Section: Introduction and Resultsmentioning

confidence: 99%

“…Namely, the expectation of the number of vertices tends to a finite limit determined by only L . This has already been pointed out in the case when

L = B^{2}

in the paper by Fodor et al ., see [10, Theorem 1.3], and in d dimensions for

L = B^{d}

by Fodor [9, Theorem 1.1].…”

Section: Introduction and Resultsmentioning

confidence: 99%

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On Random Approximations by Generalized Disc‐polygons

2020

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Random ball-polyhedra and inequalities for intrinsic volumes

Paouris

Pivovarov

2016

Monatsh Math

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We prove a randomized version of the generalized Urysohn inequality relating mean width to the other intrinsic volumes. To do this, we introduce a stochastic approximation procedure that sees each convex body K as the limit of intersections of Euclidean balls of large radii and centered at randomly chosen points. The proof depends on a new isoperimetric inequality for the intrinsic volumes of such intersections. If the centers are i.i.d. and sampled according to a bounded continuous distribution, then the extremizing measure is uniform on a Euclidean ball. If one additionally assumes that the centers have i.i.d. coordinates, then the uniform measure on a cube is the extremizer. We also discuss connections to a randomized version of the extended isoperimetric inequality and symmetrization techniques.

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Facial structure of strongly convex sets generated by random samples

Marynych

Molchanov

2022

Advances in Mathematics

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On Random Disc Polygons in Smooth Convex Discs

Cited by 9 publications

References 20 publications

On Random Approximations by Generalized Disc‐polygons

On Random Approximations by Generalized Disc‐polygons

Random ball-polyhedra and inequalities for intrinsic volumes

Facial structure of strongly convex sets generated by random samples

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