5
$\begingroup$

I got confused lately. It seems like in the metric context a polyhedron tends to mean an intersection of a finite number of half-spaces, while a polytope is a convex hull of a finite set of points. At the same time it seems that for some authors a polyhedron means something 3-dimensional, while polytopes are used for higher dimensions (Wikipedia calls 4-dimensional guys polychora). All of the above are convex, and sometimes the adjective "convex" is added, while without it finite unions of convex ones are meant.

On top of that, there is algebraic topology terminology; in that context, a polyhedron might mean something as general as a finite CW-complex. On the other hand, I found in Lurie's lectures a version for which it turns out that any open subset of a polyhedron is again a polyhedron! Here one obtains the "usual" notion if compactness is added.

So, is there some common standard terminology? What would be a canonical reference for all that?

The Related column to the right just exhibited the question non-convex Polytope definition with an answer containing a link to Grünbaum's paper with the title "Are Your Polyhedra the Same as My Polyhedra?". It begins with

"“Polyhedron” means different things to different people. There is very little in common between the meaning of the word in topology and in geometry.

That paper is from around 2003, so it seems there was no common terminology at that time. Is it still the same now?

Improve this question
$\endgroup$
6
  • 1
    $\begingroup$ It is a shame there is not an International Standards Organization that attempts to normalize mathematical terms :-). In its absence, I think you are correct that the terminology is not standardized. In many contexts, a "polyhedron" is (possibly) nonconvex unless preceded by "convex," whereas "polytope" is always convex. $\endgroup$
    – Joseph O'Rourke
    Commented Feb 9, 2017 at 12:14
  • 3
    $\begingroup$ @Joseph O'Rourke: if a polytope were always convex this book would have a different title: google.it/imgres?imgurl=http://t1.gstatic.com/… $\endgroup$
    – Qfwfq
    Commented Feb 9, 2017 at 16:07
  • 1
    $\begingroup$ @Qfwfq: I stand corrected: "polytope" is usually understood to be convex, especially more recently than 1967, when Grünbaum's book was published. $\endgroup$
    – Joseph O'Rourke
    Commented Feb 9, 2017 at 16:38
  • $\begingroup$ What do you expect as an answer? Recent mentions of polyhedron/polytopes with different meanings? Or just a simple "no"? $\endgroup$
    – Moritz Firsching
    Commented Aug 7, 2017 at 12:22
  • 1
    $\begingroup$ @MoritzFirsching The plan is to collect enough diverse input for deciding on my own terminology use $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Aug 8, 2017 at 5:13

0

Reset to default

You must log in to answer this question.