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Illumination and visibility problems in terms of closure operators. (English) Zbl 1074.52001

This paper continues the subject considered earlier by these authors in [Aequationes Math. 64, 128–135 (2002; Zbl 1013.52005)]. Recall that a point \(x\) of the boundary \(\partial (K)\) of a convex body \(K \subset R^n\) is called to be illuminated by a point \(z \not \in K\) if there exists a point \(y \in \text{ int} (K)\) such that \(x \in yz\). Moreover, we say that a set \(S\) disjoint with \(K\) illuminates \(\partial (K)\) if every point of \(\partial (K)\) is illuminated by a point of \(S\).
The authors present a few conditions equivalent to the second definition. They are expressed in terms of closure operators.

MSC:

52A01 Axiomatic and generalized convexity
52A20 Convex sets in \(n\) dimensions (including convex hypersurfaces)

Citations:

Zbl 1013.52005
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