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Abstract  

Consider a 3-dimensional point set

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which contains the incenters of all the nondegenerate tetrahedra with vertices from
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. In this paper we prove that then
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is dense in its convex hull. This settles the last unsolved variation in a sequence of similar questions initiated by D. Ismailescu, where he required to include other simplex centers, e.g. the orthocenters or the circumcenters. Our method allows us to generalize the planar incenter problem, showing that the denseness follows from a much weaker assumption for planar point sets.

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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.

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Abstract  

In this paper the following is proved: let

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be a centrally symmetric set of points, such that the distance between any pair of points is at least 1 and every three of them can be covered by a strip of width 1. Then there is a strip of width √2 covering
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.

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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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Abstract  

We say that a convex set K in ℝd strictly separates the set A from the set B if A ⊂ int(K) and B ⋂ cl K = ø. The well-known Theorem of Kirchberger states the following. If A and B are finite sets in ℝd with the property that for every TAB of cardinality at most d + 2, there is a half space strictly separating TA and TB, then there is a half space strictly separating A and B. In short, we say that the strict separation number of the family of half spaces in ℝd is d + 2. In this note we investigate the problem of strict separation of two finite sets by the family of positive homothetic (resp., similar) copies of a closed, convex set. We prove Kirchberger-type theorems for the family of positive homothets of planar convex sets and for the family of homothets of certain polyhedral sets. Moreover, we provide examples that show that, for certain convex sets, the family of positive homothets (resp., the family of similar copies) has a large strict separation number, in some cases, infinity. Finally, we examine how our results translate to the setting of non-strict separation.

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