Search Results

You are looking at 1 - 4 of 4 items for :

  • All content x
Clear All

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.

Restricted access

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.

Restricted access

Abstract  

Fix k, d, 1 ≤ kd + 1. Let

\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathcal{F}$$ \end{document}
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
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathcal{F}$$ \end{document}
}. Assume that, for every k (not necessarily distinct) members F 1, …, F k of
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathcal{F}$$ \end{document}
,∪{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
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathcal{F}$$ \end{document}
,∪{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.

Restricted access

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

\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathcal{B}$$ \end{document}
(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
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathcal{B}$$ \end{document}
(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
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathcal{B}$$ \end{document}
(V) and show that in this case the boundary complex of
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathcal{B}$$ \end{document}
(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.

Restricted access