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. (There is no need to break up the polytope into cones.)
(Compare this with my answer to an earlier question, A convex curve inside the unit circle and see the drawing in it.)