A modification of Victor Prostak's answer yields a short and almost elementary proof:

Since the set of convex $n$-dimensional convex polytopes is dense in the space of all $n$-dimensional convex bodies, it suffices to prove the inequality in question for convex polytopes, and this is quite obvious when you erect perpendicular prisms of height $\varepsilon$ based on the polytope's facets, one family outwards the polytope, the other one inwards.