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. 



non-convex Polytope definition
(This should be a comment, but don´t know how to post it as such)


The words "polytope" and "polyhedron" can mean different things... and different people mean different things when they say "regular polytope". 


Take a look at Grünbaum´s paper: 
Are your polyhedra the same as my polyhedra?, Discrete and Computational Geometry: The Goodman-Pollack Festschrift. B. Aronov, S. Basu, J. Pach, and Sharir, M., eds. Springer, New York 2003, pp. 461 – 488 (<http://www.math.washington.edu/~grunbaum/Your%20polyhedra-my%20polyhedra.pdf>)


