# Polyhedra vs Polytope

I am having a hard time understanding what is the main difference between a polyhedron and a polytope. Could anyone explain me what is the difference between these two structures?

• There is no consensus on what these terms mean. The question is not answerable. – quid Feb 19 '16 at 20:08
• @quid "These terms are used inconsistently, namely X, Y and Z" would be a good answer. – user147263 Feb 19 '16 at 20:09
• @Sally maybe, maybe I will try. – quid Feb 19 '16 at 20:11
• Can you give us the definitions you are looking at? As others have pointed out, different sources use these words differently. If we knew more about where you are seeing them, it would be easier to explain their difference in a way that suits your case. – Sean English Feb 19 '16 at 20:33
• Thanks! I am reading this in the context of tropical geometry, using the book by Sturmfels and Maclagan, "Introduction to Tropical Geometry". – user284639 Feb 19 '16 at 21:26

A polyhedron is a special case of a polytope, or, equivalently, a polytope is a generalization of a polyhedron. A polytope has a certain dimension $n$, and when $n=3$ we say that the polytope is a polyhedron. (Similarly when $n=2$ we say that the polytope is a polygon.)

This is analogous to how we can define a general $n$-dimensional sphere, and how we call the $n=1$ case a "circle".

EDIT: Indeed I should mention that this definition is not universal. Some people say "polyhedron" to mean "polytope" as I've used it above, and say "polytope" to mean "bounded polyhedron".

• @Alex Provost, what about bounded polytopes in spaces of dimensions greater than 3? Are these not called polyhedra too? – gen Jan 17 '18 at 12:49
• @gen According to both "standard" definitions mentioned in the answer above, the preferred term would be polytope and not polyhedron in that situation. – Alex Provost Jan 17 '18 at 16:12
• Standard usage in geometric combinatorics and polyhedral optimization (and this is the context in which the Maclagan-Sturmfels book is written) a polyhedron is a the solution set of a finite system of linear inequalities and a polytope is the convex hull of a finite set of points. Every polytope is a polyhedron and every bounded polyhedron is a polytope. That is: "polytope" = "bounded polyhedron". – Francisco Santos Jun 24 '18 at 9:11
• @FranciscoSantos Yes; this is the content of the last paragraph. – Alex Provost Jun 24 '18 at 17:09

In his classic text on Convex Polytopes, Grünbaum gave three incompatible definitions of a polyhedron, each used in a different branch of mathematics. There are plenty more. He later wrote somewhat resignedly that a polyhedron "means whatever you want it to mean".

Definitions of a polytope also vary, though thankfully not quite as much.

In the original sense, a polytope was an n-dimensional generalization of a 3-dimesional polyhedron or 2-dimesional polygon (whatever those are).

Some branches of mathematics find it convenient to distinguish them in kind, with say one being a bounded n-space and the other either unbounded or the bounding (n−1)-surface. Some insist on convexity, others do not.

Authors are often lax in specifying which kinds they are referring to and this sometimes leads to them incorrectly drawing on results derived for another kind. If the document you are reading is not clear what it means, be wary of its validity.