# Polyhedron, understanding face vs facet.

I've the two following definitions, for which I was trying to understand the difference.

For a given polyhedron $P$ a face $F$ is both $P$ itself or the intersection of $F$ with $P$. A facet is instead a maximal face distinct from $P$.

I don't understand in this context both "maximal" and "distinct". For a given polyhedron $P$ whatever the face we are considering is, isn't such face distinct from $P$ by definition? Also if the the face is described by a single linear equation what's the meaning of "maximal" in this context?

Update:

I want to quote the exact definitions I have for both face and facet. Face:

A face of $P$ is $P$ itself or the intersection of $P$ with a supporting hyperplane of $P$

Facet:

A facet of $P$ is a maximal face distinct from $P$. An inequality $cx \leq \delta$ is facet-defining for $P$ if $cx \leq \delta$ for all $x \in P$ and $\left\{x \in P : cx = \delta \right\}$ is a facet of $P$

In the context $P$ is a polyheadron, $cx$ is a dot product, both $c$ and $x$ are vectors. I don't understand what "maximal" means in this context. Any chance that maximal means greatest dimension?

Update 2:

I'm still puzzled, but it's probably me that I cannot absorb these definitions. If $c$ is a non zero vector for which $\delta = max \left\{ cx : x \in P \right\}$ is finite then the set $\left\{ x \;:\; cx = \delta \right\}$ is called supporting hyperplane.

So a face, by definition is a subset of $P$, and then it's a polyhedron itself, so it make sense to compute the dimension. Let $n$ be the dimension of $P$, If specifically a supporting hyperplane $H$ is not $P$ then the dimension is at most $n-1$ (is it right? I'm not sure here). A facet is a maximal face distinct from $P$, since the dimension can be at most $n-1$ then a facet is any face of such dimension, is that right?

• I'd really like to know from which book I can learn the definition of face and facet. – Mark Dec 26 '18 at 7:20
• @Mark, chapter 3 of books.google.it/… – user8469759 Dec 26 '18 at 13:59

A facet is just a special type of face.

According to Wikipedia:

The facets of an $n-$polytope are the faces of the polytope with dimension $n-1$.

However a face can have many more dimensions. So, for example, given a 3-dimensional cube, not only are the 2-dimensional facets (the squares which are the sides of the cube) are faces, but also the edges are 1-dimensional faces, and the vertices are 0-dimensional faces.

Obviously in this regard the use of the term "face" differs from that used in elementary geometry, where only the facets would be considered faces. This is perhaps the source of your confusion.

https://en.wikipedia.org/wiki/Face_(geometry)#Facet_or_.28n-1.29-face

• What you said gives me the idea, however now I would like to understand if the definitions I gave is equivalent to this descrpition or not. – user8469759 Jun 28 '16 at 8:21
• They look equivalent to me -- where exactly are you confused. The first definition says that a face can have any dimension from $0$ (a vertex) to $n$ (the polytope itself), and it is clear that any face besides the polytope itself is just the intersection of the polytope with a supporting hyperplane. When they say "maximal" face, they mean "maximal" dimension. Clearly the largest possible dimension for any face besides the polytope itself must be $n-1$, and that is exactly what the definition from Wikipedia says. As you said yourself, maximal does mean greatest dimension. – Chill2Macht Jun 28 '16 at 13:08
• I don't know, it just looks twisted to me (the definitions I gave). I update my original post with I've understood so far. – user8469759 Jun 28 '16 at 13:15
• I edited, can you please tell me if what I said is correct or not? – user8469759 Jun 28 '16 at 13:23
• The confusion came up probably because it doesn't look like what for maximal is meant (for the record the book is this one: books.google.it/…) – user8469759 Jun 28 '16 at 13:30