I have a simple question. I read that given a vector space $N_{\mathbb{R}}$ over $\mathbb{R}$, we can define a convex polytope in the following way:

$$P:= \Big\{ \sum_{u\in S} \mu_u u \,\Big| \, \mu_u \geq 0 , \sum_{u\in S} \mu_u =1 \Big\} \subset N_{\mathbb{R}}$$

with $S$ finite.

What is the definition of polytope and regular polytope in general?

Thanks in advance.

regularityshould mean in Coxeter's bookRegular polytopes. $\endgroup$ – Mariano Suárez-Álvarez May 11 '13 at 20:12polytopal complex? This is defined in Ziegler'sLectures on Polytopes. $\endgroup$ – Joseph O'Rourke May 12 '13 at 0:54