[EDIT: The following is my own clumsy mistake] Minor note: The definition of "mean width" of a polyhedron P in the paper is not translation invariant, and that's confusing. In other words, the mean width of a polyhedron P can differ from that of P+x := {p+x | p ∈ P} where x is some vector. Is that intended? It doesn't agree with how the word "width" is normally used. I would call it a "mean furthest projection". Or maybe "mean peak projection" or "mean shadow"?
"Half the mean width of a polyhedron P is equal to the expected value of
max θ^T x
subject to x ∈ P,
The expression max θ^T x is not translation-invariant: if you replace x with x + ∆x, you get (max θ^T x) + θ^T ∆x. But the expectation of θ^T ∆x is 0 so the expectation of the maximum is translation-invariant again.