Ball and spindle convexity with respect to a convex body

Lángi, Zsolt; Naszódi, Márton; Talata, István

doi:10.1007/s00010-012-0160-z

Cited by 19 publications

(16 citation statements)

References 30 publications

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“…The notion of spindle convexity can further be generalized by replacing the radius r circular disc by a fixed convex disc L . This leads to the notions of L ‐convexity and L ‐spindle convexity, as introduced in [16]. For a historical overview of this topic and references, see the introduction of [16].…”

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: 99%

“…Let K and L be convex discs (to avoid technical complications we always assume that the sets involved are compact). We say that K is L ‐convex [16, Definition 1.1] if it is equal to the intersection of all translates of L that contain K . Of course, if K is L ‐convex, then it is also convex in the usual sense.…”

Section: Introduction and Resultsmentioning

confidence: 99%

On Random Approximations by Generalized Disc‐polygons

2020

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“…where H is a largest area affine-regular hexagon inscribed in M . Thus, λ(M ) ≥ λ(H) implies that (12) c m * tr (K) ≥ 8 (λ(M ) + 3λ(P )) 3λ(P ) .…”

Section: The Proofs Of 21 23 and 24mentioning

confidence: 99%

“…For any K ∈ K n , we say that the relative norm of K is the norm with the central symmetral 1 2 (K − K) of K as its unit ball (cf. [12] or [11]). Observe that, up to multiplication by a scalar, the relative norm of K is the unique norm in which K is a body of constant width.…”

Section: Introductionmentioning

confidence: 99%

On a normed version of a Rogers–Shephard type problem

Lángi

2016

Isr. J. Math.

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A translation body of a convex body is the convex hull of two of its translates intersecting each other. In the 1950s, Rogers and Shephard found the extremal values, over the family of n-dimensional convex bodies, of the maximal volume of the translation bodies of a given convex body. In our paper, we introduce a normed version of this problem, and for the planar case, determine the corresponding quantities for the four types of volumes regularly used in the literature: Busemann, Holmes-Thompson, and Gromov's mass and mass*. We examine the problem also for higher dimensions, and for centrally symmetric convex bodies.1991 Mathematics Subject Classification. 52A38, 52A21, 52A40.

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“…Remark We note that r ‐ball bodies and r ‐ball polyhedra have been intensively studied (under various names) from the point of view of convex and discrete geometry in a number of publications (see the recent papers , and the references mentioned there).…”

Section: Introductionmentioning

confidence: 99%

On Uniform Contractions of Balls in Minkowski Spaces

Bezdek

2020

Mathematika

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Let N balls of the same radius be given in a d-dimensional real normed vector space, i.e., in a Minkowski d-space. Then apply a uniform contraction to the centers of the N balls without changing the common radius. Here a uniform contraction is a contraction where all the pairwise distances in the first set of centers are larger than all the pairwise distances in the second set of centers. The main results of this paper state that a uniform contraction of the centers does not increase (respectively, decrease) the volume of the union (respectively, intersection) of N balls in Minkowski d-space, provided that N 2 d (respectively, N 3 d and the unit ball of the Minkowski d-space is a generating set). Some improvements are presented in Euclidean spaces. §1. Introduction. The Kneser-Poulsen conjecture [17,26] (respectively, Gromov-Klee-Wagon conjecture [12,15,16]) states that if the centers of a family of N unit balls in Euclidean d-space is contracted, then the volume of the union (respectively, intersection) does not increase (respectively, decrease). These conjectures have been proved by Bezdek and Connelly [5] for d = 2 (in fact, for not necessarily congruent circular disks as well) and they are open for all d 3. For a number of partial results in dimensions d 3, we refer the interested reader to the corresponding chapter in [3]. Very recently Bezdek and Naszódi [8] investigated the Kneser-Poulsen conjecture as well as the Gromov-Klee-Wagon conjecture for special contractions in particular, for uniform contractions. Here, a uniform contraction is a contraction where all the pairwise distances in the first set of centers are larger than all the pairwise distances in the second set of centers. The main result of [8] states that a uniform contraction of the centers does not increase (respectively, decrease) the volume of the union (respectively, intersection) of N unit balls in Euclidean d-space (d 3), provided that N c d d 2.5d , where c > 0 is a universal constant (respectively, N (1 + √ 2) d ). In this paper we improve these results and extend them to Minkowski spaces.Let K ⊂ R d be an o-symmetric convex body, i.e., a compact convex set with non-empty interior symmetric about the origin o in R d . Let · K denote the norm generated by K, which is defined by x K := min{λ 0 | λx ∈ K} for x ∈ R d . Furthermore, let us denote R d with the norm · K by M d K and call it the Minkowski space of dimension d generated by K. We write B K [x, r] := x + rK for x ∈ R d and r > 0 and call any such set a (closed) ball of radius r, where + refers to vector addition extended to subsets of R d in the usual way. The following definitions introduce the core notions and notations for our paper. Definition 1. For X ⊆ R d and r > 0 let X K r := {B K [x, r] | x ∈ X } respectively, X r K := {B K [x, r] | x ∈ X }

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Ball and spindle convexity with respect to a convex body

Cited by 19 publications

References 30 publications

On Random Approximations by Generalized Disc‐polygons

On Random Approximations by Generalized Disc‐polygons

On a normed version of a Rogers–Shephard type problem

On Uniform Contractions of Balls in Minkowski Spaces

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