Definition 1.6 of Delaunay Triangulation - Curve and Surface Reconstruction

Question

I'm reading the very first chapter of Tamal K. Dey - Curva and Surface Reconstruction.

There's a small notation issue I don't quite get in the following definition:

Definition 1.6 : The Delaunay Triangulation of $P$ is a simplicial complex $$ \text{Del}\;P = \left\{ \sigma = \text{Conv}\;T : \bigcap_{p\in T \subset P} V_p \neq \emptyset \right\} $$

Here $V_p$ is a Voronoi cell. I'm not sure what $\text{Conv} \; T$ is supposed to be. If I go back the only place where I see such notation used is in the following:

A simplex $\sigma' = \text{Conv} \; T$ for a nonempty subset $T \subset P$ is called a face of $\sigma$. $\sigma$ is a k-simplex and $P$ is a set of points in $\mathbb{R}^k$.

I'm tempted to think that $\text{Conv} \; T$ is a face of the $k$-simplex, but it there's no actual definition and I don't want to misunderstand it.

Answer 1

By '$\text{Conv}~T$', the author certainly mean the convex hull or convex combination. For k+1 point in dimension k, it is just a simplex (triangle in 2D, tetrahedron in 3D...)

The mathematical definition is described in the books with a sentence:

In words, k+1 points in P form a Delaunay k-simplex if their Voronoi cells have nonempty intersection.

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I however agree that both are difficult to understand, so I will explain it with my own words:

If you can find a set of k+1 Voronoi cells that all intersect (every pair has an intersection), then the points associated with the Voronoi cells form a k-simplex (a triangle in 2D) of the Delaunay triangulation.

For example, in the following image, select 3 Voronoi cells at random. If they have at least a point in common, there is a triangle of the Delaunay triangulation that connects the points (called seeds, sites, or generators) of those same Voronoi cells.