I assume you're talking about this? "Half the mean width of a polyhedron P is equal to the expected value of max θ^T x subject to x ∈ P, where θ ∈ S^(d−1

I assume you're talking about this? "Half the mean width of a polyhedron P is equal to the expected value of max θ^T x subject to x ∈ P, where θ ∈ S^(d−1) is uniformly random distributed with respect to the Haar measure on the unit sphere." 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.