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?