In a polytope, what are the difference and relation between facet and face? How are they defined respectively? Thanks and regards!
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Usually facet is synonymous with maximal face. Or in other words, if the polytope is of dimension $d$, the facets are the faces of dimension $d-1$, or codimension $1$.
A face is just a common name for $\emptyset$, vertices, edges, and so on. Often one says that a $k$-dimensional face is called an "$n$-face". Usually one also says that the whole polytope is a face also (this is to ensure that intersection of faces is also a face). The mathematical definition of a face varies in the literature (as the Wikipedia article mentions) - but often one says that a face of a polytope is a subset of the polytope maximizing some linear functional (though this definition is not very intuitive...)