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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...)

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(1) What are faces of a polytope then? (2) How would you call a 2-dimensinal face? Thanks, +1! –  Tim Dec 15 '12 at 18:20
    
@Tim: I've added some more on faces also. –  Fredrik Meyer Dec 15 '12 at 18:25
    
Thanks! (1) Is there a name for a 2-dimensinal face? (2) Also what is "some linear functional"? –  Tim Dec 15 '12 at 18:31
    
I see that a 2D face is called a face! –  Tim Dec 15 '12 at 18:50

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