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Properties of \(n\)-dimensional triangulations. (English) Zbl 0624.65018

This paper establishes a number of mathematical results relevant to the problem of constructing a triangulation, i.e., a simplicial tesselation of the convex hull of an arbitrary finite set of points in n-space. The principal results of the present paper are (a) A set of \(n+2\) points in n-space may be triangulated in at most 2 different ways. (b) The ‘sphere test’ defined in this paper selects a preferred one of these two triangulations. (c) A set of parameters is defined that permits the characterization and enumeration of all sets of \(n+2\) points in n-space that are significantly different from the point of view of their possible triangulations. (d) The local sphere test induces a global sphere test property for a triangulation. (e) A triangulation satisfying the global sphere property is dual to the n-dimensional Dirichlet tesselation, i.e., it is a Delaunay triangulation.

MSC:

65D99 Numerical approximation and computational geometry (primarily algorithms)
05B45 Combinatorial aspects of tessellation and tiling problems
52C17 Packing and covering in \(n\) dimensions (aspects of discrete geometry)
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