The article is devoted to generalization of Delaunay triangulation. We suggest to consider empty condition for special convex sets. For given finite set $P\subset \mathbb{R}^n$ we shall say that empty condition for convex set $B\subset \mathbb{R}^n$ is fullfiled if $P\cap B=P\cap \partial B$. Let $\Phi=\Phi_{\alpha}, \alpha \in {\mathcal A}$ be a family of compact convex sets with non empty inner. Consider some nondegenerate simplex $S\subset \mathbb{R}^n$ with vertexes $p_0,...,p_n$. We define the girth set $B(S)\in \Phi$ if $q_i\in \partial B(S), i=0,1,...,n$. We suppose that the family $\Phi$ 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 $\Phi=\Phi_{\alpha}, \alpha\in {\mathcal A}$ of convex sets have the pointed above property then for the girth sets it is true:
The set $B(S)$ is uniquely determined by any simplex with vertexes on $\partial B(S)$.
Let $S_1,S_2$ be two nondegenerate simplexes such that $B(S_1)\ne B(S_2)$. If the intersection $B(S_1)\cap B(S_2)$ is not empty, then the intersection of boundaries $B(S_1), B(S_2)$ is $(n-2)$-dimensional convex surface, lying in some hyperplane.
If two simplexes $S_1$ and $S_2$ 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 $B(S_2)$ such that $B\not \in B(S_1)$ then $B(S_2)$ does not contain the vertex $A$ of simplex $S_1$.
These statements allow us to define $\Phi$-triangulation correctly by the following way. The given triangulation $T$ of finite set $P\subset \mathbb{R}^n$ is called $\Phi$-triangulation if for all simlex $S\in T$ the girth set $B(S)\in Phi$ is empty.
In the paper we give algorithm for construct $\Phi$-triangulation arbitrary finite set $P\subset \mathbb{R}^n$.
Besides we describe exapmles of families $\Phi$ for which we prove the existence and uniqueness of girth set $B(S)$ for arbitrary nondegenerate simplex $S$.
