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,

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.