TITLE: Definition of simplex
QUESTION [14 upvotes]: From Wikipedia:

an n-simplex is an n-dimensional
  polytope which is the convex hull of
  its n + 1 vertices.

I was wondering if the definition is equivalent to say a simplex is synonym of a convex polytope?
Is simplex defined only for $\mathbb{R}^n$, not for other more general spaces?
Thanks and regards!

REPLY [6 votes]: I often work over simplices in $\mathbb{F}_q^d$ or other discrete spaces.  Here, when we say a $k$-simplex, we mean a set of $(k+1)$-points spanning a $k$-dimensional subspace.  I.e., for us, a $k$-simplex is just the set of vertices.