Ball convex bodies in Minkowski spaces

Abstract

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.

Variance estimates for random disc-polygons in smooth convex discs

“…It is known (see [29,Section 3.2]) that in this case a closed circular disc of radius r m = 1/κ M rolls freely in K, that is, for each x ∈ ∂K, there exists a p ∈ R 2 with x ∈ r m B 2 + p ⊂ K. Moreover, K slides freely in a circle of radius r M = 1/κ m , which means that for each x ∈ ∂K there is a vector p ∈ R 2 such that x ∈ r M ∂B 2 + p and K ⊂ r M B 2 + p. The latter yields that for any two points x, y ∈ K, the intersection of all closed circular discs of radius r ≥ r M containing x and y, denoted by [x, y] r and called the r-spindle of x and y, is also contained in K. Furthermore, for any X ⊂ K, the intersection of all radius r ≥ r M circles containing X, called the closed r-hyperconvex hull (or r-hull for short) and denoted by conv r (X), is contained in K. The concept of hyperconvexity, also called spindle convexity or r-convexity, can be traced back to Mayer [21]. For a systematic treatment of geometric properties of hyperconvex sets and further references, see, for example, [10] and [19], and in a more general setting [20]. The notion of convexity arises naturally in many questions where a convex set can be represented as the intersection of equal radius closed balls.…”

Facial structure of strongly convex sets generated by random samples

“…If K is the Euclidean ball, then bh K (A) is called the ball hull of A and a K-strongly convex set is called ball convex, see [3,4]. If K ∈ K d (0) is (origin) symmetric, that is, K = −K, the K-hull can be viewed as the ball hull in the Minkowski space with K being its unit ball, see [13].…”

“…If K is not necessarily a generating set, the proof follows the scheme of the proof of this fact for origin symmetric K in [13]. Assume that the segment conv…”

“…This concept (called the K-strong convexity) has been intensively studied in [1,24] and accompanying works. If K is origin symmetric, it can be considered as the unit ball in a Minkowski space and the K-hull of A becomes the ball hull of A in a Minkowski space, see [13] and [20], the latter also includes the case of a not necessarily origin symmetric K.…”

On the intrinsic volumes of intersections of congruent balls

“…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 .…”