## Klyachin V.A. Triangulation Algorithm Based on Empty Convex Set Condition

http://dx.doi.org/10.15688/jvolsu1.2015.3.3

Doctor of Physical and Mathematical Sciences,
Head of Department of Computer Science and Experimental Mathematics,
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Prosp. Universitetsky, 100, 400062 Volgograd, Russian Federation

Abstract. The article is devoted to generalization of Delaunay triangulation. We suggest to consider empty condition for special convex sets. For given finite set P ⊂ Rn we shall say that empty condition for convex set B ⊂ Rn is fullfiled if P ∩ B = P ∩ ∂B. Let Φ = Φα, α ∈ A  be a family of compact convex sets with non empty inner. Consider some nondegenerate simplex S ⊂ Rn with vertexes p0,...,pnWe define the girth set B(S) ∈ Φ if qi ∈ B(S), i = 0, 1, ..., n. We  suppose that the family Φ has the property: for arbitrary nondegenerate simplex S there is only one the girth set B(S). We prove the following main result.
Theorem 1. If the family Φ = Φα, α A of convex sets have the pointed above property then for the girth sets it is true:
1. The set B(S) is uniquely determined by any simplex with vertexes on B(S).
2. Let S1, S2 be two nondegenerate simplexes such that B(S1) ≠ B(S2). If the intersection B(S1 B(S2) is not empty, then the intersection of boundaries B(S1), B(S2) is (n − 2)-dimensional convex surface, lying in some hyperplane.
3. If two simplexes S1 and S2 don’t intersect by inner points and have common (n − 1)-dimensional face G and A, B are vertexes don’t belong to face G and vertex B of simplex B(S2) such that B ∉ B(S1) then B(S2) does not contain the vertex A of simplex S1.
These statements allow us to define Φ-triangulation correctly by the following way. The given triangulation T of finite set P ⊂ Rn is called Φ-triangulation if for all simlex S ∈ T the girth set B(S) ∈ Φ is empty. In the paper we give algorithm for construct Φ-triangulation arbitrary finite set P ⊂ Rn. Besides we describe examples of families Φ for which we prove the existence and uniqueness of girth set B(S) for arbitrary nondegenerate simplex S.

Key words: triangulation, empty shpere condition, Delaunay triangulation, convex set, convex function, convex hull.