What is simplex?

A subset $P$ of $R^d$ is called a $k$-simplex ( $k =0,1, 2, \ldots$) if it is the convex hull of $k+1$ affinely independent points. It has exactly $k+1$ vertices and $k+1$ facets. A simplex is a $k$-simplex for some $k$.

Simplices are selfdual, i.e. a dual (see 2.4) of a simplex is again a simplex.