I have several questions concerning the Delaunay triangulation of a high dimensional lattice. Given an $n$-dimensional lattice $L$ and its Delaunay triangulation (partition of $R^n$ into simplices such that the circumsphere of any simplex does not contain any lattice point).

1. Does the center of a circumsphere of a Delaunay simplex have to be contained inside the simplex? 2. Does a set of any $n$ independent vectors such that each vector coincides with a Delaunay edge form a basis of $L$? 3. Is there a basis $B = \{b_1,b_2,\ldots , b_n\}$ of $L$ such that the $n$ dimensional parallelepiped $B[0,1)^n$ can be decomposed into Delaunay simplices?

Any partial answer for any question or pointing to future reading would be greatly appreciated.