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Clear All  # Generating the kernel of a staircase starshaped set from certain staircase convex subsets

Periodica Mathematica Hungarica
Author: Marilyn Breen

## Abstract

Let S be an orthogonal polygon in the plane. Assume that S is starshaped via staircase paths, and let K be any component of Ker S, the staircase kernel of S, where KS. For every x in S\K, define W K(x) = {s: s lies on some staircase path in S from x to a point of K}. There is a minimal (finite) collection W(K) of W K(x) sets whose union is S. Further, each set W K(x) may be associated with a finite family U K(x) of staircase convex subsets, each containing x and K, with ∪{U: U in U K(x)} = W K(x). If W(K) = {W K(x 1), ..., W K(x n)}, then KV K ≡ ∩{U: U in some family U K(x i), 1 ≤ in} ⊆ Ker S. It follows that each set V K is staircase convex and ∪{V k: K a component of Ker S} = Ker S. Finally, if S is simply connected, then Ker S has exactly one component K, each set W K(x i) is staircase convex, 1 ≤ in, and ∩{W k(x i): 1 ≤ i ≤ n} = Ker S.

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# A Krasnosel’skii-type result for planar sets starshaped via orthogonally convex paths

Periodica Mathematica Hungarica
Author: Marilyn Breen

## Abstract

A Krasnosel’skii-type theorem for compact sets that are starshaped via staircase paths may be extended to compact sets that are starshaped via orthogonally convex paths: Let S be a nonempty compact planar set having connected complement. If every two points of S are visible via orthogonally convex paths from a common point of S, then S is starshaped via orthogonally convex paths. Moreover, the associated kernel Ker S has the expected property that every two of its points are joined in Ker S by an orthogonally convex path. If S is an arbitrary nonempty planar set that is starshaped via orthogonally convex paths, then for each component C of Ker S, every two of points of C are joined in C by an orthogonally convex path.

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# Analogues of Horn’s theorem for finite unions of starshaped sets in ℝ d

Periodica Mathematica Hungarica
Author: Marilyn Breen

## Abstract

Fix k, d, 1 ≤ kd + 1. Let

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be a nonempty, finite family of closed sets in ℝd, and let L be a (dk + 1)-dimensional flat in ℝd. The following results hold for the set T ≡ ∪{F: F in
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}. Assume that, for every k (not necessarily distinct) members F 1, …, F k of
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,∪{F i: 1 ≤ ik} is starshaped and the corresponding kernel contains a translate of L. Then T is starshaped, and its kernel also contains a translate of L. Assume that, for every k (not necessarily distinct) members F 1, …, F k of
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,∪{F i: 1 ≤ ik} is starshaped and there is a translate of L meeting each set ker F i, 1 ≤ ik − 1. Then there is a translate L 0 of L such that every point of T sees via T some point of L 0. If k = 2 or d = 2, improved results hold.

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# Ball polytopes and the Vázsonyi problem

Acta Mathematica Hungarica
Authors: Y. Kupitz, H. Martini, and M. Perles

## Abstract

Let V be a finite set of points in the Euclidean d-space (d ≧ 2). The intersection of all unit balls B(υ, 1) centered at υ, where υ ranges over V, henceforth denoted by

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(V) is the ball polytope associated with V. After some preparatory discussion on spherical convexity and spindle convexity, the paper focuses on two central themes. (a) Define the boundary complex of
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(V), i.e., define its vertices, edges and facets in dimension 3, and investigate its basic properties. (b) Apply results of this investigation to characterize finite sets of diameter 1 in the (Euclidean) 3-space for which the diameter is attained a maximal number of times as a segment (of length 1) with both endpoints in V. A basic result for such a characterization goes back to Grünbaum, Heppes and Straszewicz, who proved independently of each other, in the late 1950’s by means of ball polytopes, that the diameter of V is attained at most 2|V| − 2 times. Call V extremal if its diameter is attained this maximal number (2|V| − 2) of times. We extend the aforementioned result by showing that V is extremal iff V coincides with the set of vertices of its ball polytope
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(V) and show that in this case the boundary complex of
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(V) is self-dual in some strong sense. The problem of constructing new types of extremal configurations is not addressed in this paper, but we do present here some such new types.

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