non-convex Polytope definition 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. 
 A: (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)
A: One possible definition is that a concrete polytope is an abstract polytope where each vertex has been assigned to a point in space, and all elements of rank n are contained in nD subspaces.
A: The pentagram either can be considered as a sequence of 5 sides circling around the center twice, or as a stellation of the central pentagon. The same applies to the halfspace generations of polytopes of any dimension too. Just cf. to


*

*https://en.wikipedia.org/wiki/Winding_number

*https://en.wikipedia.org/wiki/Density_(polytope)

*https://en.wikipedia.org/wiki/Stellation
--- rk
