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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.

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  • $\begingroup$ Apparently wikipedia has a very broad definition of polytope en.wikipedia.org/wiki/Polytope But I have only seen polytope be used to mean that definition, which immediately implies that it is convex $\endgroup$
    – David Benson-Putnins
    Commented May 11, 2013 at 20:07
  • $\begingroup$ There are various definitions of what regularity should mean in Coxeter's book Regular polytopes. $\endgroup$
    – Mariano Suárez-Álvarez
    Commented May 11, 2013 at 20:12
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    $\begingroup$ The title of your question ("non-convex") does not match the question itself. Perhaps you are seeking a definition of a polytopal complex? This is defined in Ziegler's Lectures on Polytopes. $\endgroup$
    – Joseph O'Rourke
    Commented May 12, 2013 at 0:54

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(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)

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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.

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  • $\begingroup$ This is nearly my preferred definition. I would say that an (abstract) element of rank $n$ is assigned to (rather than contained in) an $n$-dimensional subspace. There's no reference to a particular subset of this subspace; this eliminates problems with "holes" like the centre of a star polytope. -- You could also provide a definition or link for abstract polytopes. $\endgroup$
    – mr_e_man
    Commented May 4, 2022 at 2:40
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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

--- rk

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