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## 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

*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 *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*.