Intersections of Balls and Sets of Constant Width in Finite‐dimensional Normed Spaces

Martini, Horst; Richter, Christian; Spirova, Margarita

doi:10.1112/s0025579313000041

Cited by 10 publications

(15 citation statements)

References 17 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…holds for any supporting r-ball B of conv r ∆. Finally, we prove (19) as follows. First, we note that clearly,…”

Section: Introductionmentioning

confidence: 91%

“…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 [2], [14], [16], [17], [19], and the references mentioned there). In particular, the following Blaschke-Santaló-type inequalities have been proved by Paouris and Pivovarov (Theorem 3.1 in [20]) as well as the author (Theorem 1 in [7]) for r-ball bodies in E d .…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On the intrinsic volumes of intersections of congruent balls

Bezdek

2022

Discrete Optimization

View full text Add to dashboard Cite

We investigate the intersections of balls of radius r, called r-ball bodies, in Euclidean d-space. An r-lense (resp., r-spindle) is the intersection of two balls of radius r (resp., balls of radius r containing a given pair of points). We prove that among r-ball bodies of given volume, the r-lense (resp., r-spindle) has the smallest inradius (resp., largest circumradius). In general, we upper (resp., lower) bound the intrinsic volumes of r-ball bodies of given inradius (resp., circumradius). This complements and extends some earlier results on volumetric estimates for r-ball bodies. * Keywords and phrases: Euclidean d-space, r-ball body, r-ball polyhedron, intrinsic volume, inradius, circumradius, r-lense, r-spindle.

show abstract

“…holds for any supporting r-ball B of conv r ∆. Finally, we prove (19) as follows. First, we note that clearly,…”

Section: Introductionmentioning

confidence: 91%

Section: Introductionmentioning

confidence: 99%

On the intrinsic volumes of intersections of congruent balls

Bezdek

2022

Discrete Optimization

View full text Add to dashboard Cite

show abstract

“…For a nontrivial bounded, closed, and convex set K in a normed linear space X, η (K) is the union of all completions of K, and θ (K) is the intersection of all completions of K. Moreover, K has a unique completion if and only if δ(η (K)) = δ(K). See, e.g., [3] and, for the case of Minkowski spaces, [10].…”

Section: Ifmentioning

confidence: 99%

Uniqueness of completions and related topics

Martini

2018

Arch. Math.

Self Cite

View full text Add to dashboard Cite

A bounded subset of a normed linear space is said to be (diametrically) complete if it cannot be enlarged without increasing the diameter. A complete super set of a bounded set K having the same diameter as K is called a completion of K. In general, a bounded set may have different completions. We study normed linear spaces having the property that there exists a nontrivial segment with a unique completion. It turns out that this property is strictly weaker than the property that each complete set is a ball, and it is strictly stronger than the property that each set of constant width is a ball. Extensions of this property are also discussed.

show abstract

“…Suppose that this is not the case; i.e., there exists z 0 ∈ F 0 ∩ cl(S). The inclusions (31) and (32) yield z 0 − y 0 ≤ 1 and z 0 − y 1 < 1. Finally, the inclusion z 0 ∈ F 0 ⊆ S λ0 = S(y 0 + λ 0 (y 1 − y 0 ), 1) gives…”

Section: Minimal Representation Of Ball Convex Bodies As Ball Hullsmentioning

confidence: 99%

Ball convex bodies in Minkowski spaces

2017

Self Cite

View full text Add to dashboard Cite

Abstract. The notion of ball convexity, considered in finite dimensional real Banach spaces, is a natural and useful extension of usual convexity; one replaces intersections of half-spaces by suitable intersections of balls. A subset S of a normed space is called ball convex if it coincides with its ball hull, which is obtained as intersection of all balls (of fixed radius) containing S. Ball convex sets are closely related to notions like ball polytopes, complete sets, bodies of constant width, and spindle convexity. We will study geometric properties of ball convex bodies in normed spaces, for example deriving separation theorems, characterizations of strictly convex norms, and an application to complete sets. Our main results refer to minimal representations of ball convex bodies in terms of their ball exposed faces, to representations of ball hulls of sets via unions of ball hulls of finite subsets, and to ball convexity of increasing unions of ball convex bodies.

show abstract

Intersections of Balls and Sets of Constant Width in Finite‐dimensional Normed Spaces

Cited by 10 publications

References 17 publications

On the intrinsic volumes of intersections of congruent balls

On the intrinsic volumes of intersections of congruent balls

Uniqueness of completions and related topics

Ball convex bodies in Minkowski spaces

Contact Info

Product

Resources

About