What is the rigorous definition of a polyhedron and polytopes more generally?

Question

What is the formal, rigorous definition of a polyhedron in 3d space, and polytopes more generally in N-dimensional space? And, although this was not part of the title question, I also want a definition of polyhedral region and polytopal regions, that is, the region in 3d (respectively N-dimensional) that is bounded by a polyhedron (respectively polytope). I am interested in this question because of Euler's polyhedron theorem relating vertices, edges, and faces. I know that Euler's theorem fails for polyhedrons with holes in them. I would like a rigorous definition of vertex, edge, and face, as well as their higher-dimensional analogs in higher-dimensional space. Also, is there a book or paper that rigorously defines these notions?

Answer 1

EDIT: I clearly did not read the question carefully. I’ll leave this up for the special case that the polytope is convex, but this answer is badly incomplete for the general nonconvex polyhedra.

For a reference, I would recommend Ziegler's Lectures on Polytopes. It has some nice pictures.

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There are two common and equivalent definitions of a polytope.

A polytope is either:

• The convex hull of finitely many points. Here, the convex hull of a set $S$, written $\text{conv}(S)$, is the set $$\text{conv}(S) = \left\{\sum_{i=1}^n \lambda_i s_i : s_i \in S \text{ and }\sum_{i=1}^n \lambda_i = 1\right\}$$ of all convex combinations of $S$. Equivalently, it is the smallest convex set which contains $S$. Draw some pictures to convince yourself that the convex hull of finitely many points in $\mathbf{R}^2$ is a polygon, and a polyhedron in $\mathbf{R}^3$.

• The bounded intersection of finitely many half spaces. Here, a half space is everything to one side of a hyperplane, which can be written (if the hyperplane does not go through the origin) as $$H^a_- = \{x \in \mathbf{R}^d : \langle x, a\rangle \leq 1\}$$ where the $a$ in the superscript denotes $H$'s normal vector, and the $-$ in the subscript denotes the $\leq$. Draw some pictures to convince yourself that a half space in $\mathbf{R}^2$ is what you get when you draw a line, and then take all the points to one side of that line. One you feel comfortable with that, observe that a convex polygon with $n$ sides is a bounded intersection of $n$ half spaces, after removing redundant half spaces.

A polyhedron is the intersection of finitely many half spaces, and as such might be unbounded. One example of a polyhedron is $$\bigcap_{i=1}^n \{x \in \mathbf{R}^d : x_i \geq 0\}$$ the positive orthant.

For high-dimensional polytopes, there are too many dimensions to give unique names to each of the "face-like" things, so we just call all of them faces. Define a face $F$ of a polytope $P$ to be a set which can be isolated by a the boundary of a half space; i.e., there is a vector $a$ so that $$H_-^a \cap P = P \text{ and } \partial H_-^a \cap P = F$$ Draw a picture and verify that a face of a polygon is either a vertex or an edge, and a face of a polyhedron is a face (in the sense of polyhedra), an edge, or a vertex.

The distinguished faces we like are vertices, which is a face of cardinality one, and facets, which is the lingo for faces that are $d-1$-dimensional polytopes. A facet of a polygon is an edge, and a facet of a polyhedron is a face.