## 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 *K* ≠ *S*. 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 *K* ⊆ *V*
_{K} ≡ ∩{*U*: *U* in some family *U*
_{K}(*x*
_{i}), 1 ≤ *i* ≤ *n*} ⊆ 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 ≤ *i* ≤ *n*, and ∩{*W*
_{k}(*x*
_{i}): 1 ≤ i ≤ *n*} = Ker *S*.

## 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.

## Abstract

Fix *k, d*, 1 ≤ *k* ≤ *d* + 1. Let

^{d}, and let

*L*be a (

*d*−

*k*+ 1)-dimensional flat in ℝ

^{d}. The following results hold for the set

*T*≡ ∪{

*F*:

*F*in

*k*(not necessarily distinct) members

*F*

_{1}, …,

*F*

_{k}of

*F*

_{i}: 1 ≤

*i*≤

*k*} 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

*F*

_{i}: 1 ≤

*i*≤

*k*} is starshaped and there is a translate of

*L*meeting each set ker

*F*

_{i}, 1 ≤

*i*≤

*k*− 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.

## 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

*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

*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

*V*) and show that in this case the boundary complex of

*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.