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](http://www.math.harvard.edu/~lurie/287xnotes/Lecture17.pdf) 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 https://mathoverflow.net/q/130363/41291 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?