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?
Best Answer
A facet is just a special type of face.
According to Wikipedia:
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