Abstract
Given a set of points S having no accumulation points contained in a subset T of the plane, associate with each element a of S the subset Sa of points in T closer to a than the other points of S. The subsets Sa are convex polygons and partition T. T is not necessarily closed or bounded, but of course, if T is compact then S is finite. These polygons are variously referred to as Dirichlet cells, Meijering cells (1953), Voronoi regions (1908), Gilbert cells (1961), cells or tiles, and the partition is called an S-mosaic or mosaic. If the elements of S are sprinkled randomly in T call the partition a random S-mosaic. The point a is called an S-point for the tile Sa. Generalizations of this idea to higher dimensional spaces (even to metric spaces) are apparent. We have not looked into any of these in detail, although metric space (using just metric convexity?) or normed linear space generalizations of known En results might be interesting.
The research of this author was supported by NRC Grant No.5210
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Lewis, J.E., Rogers, T. (1974). Some Remarks on Random Sets Mosaics. In: van den Driessche, P. (eds) Mathematical Problems in Biology. Lecture Notes in Biomathematics, vol 2. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-45455-4_20
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DOI: https://doi.org/10.1007/978-3-642-45455-4_20
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