If $K\subset \mathbb{R}^n$ is a convex body containing the origin, let $K'$ be the convex hull of $K$ and $-K$. One of Rogers-Shephard inequalities asserts:
$$\operatorname{vol}(K') \le 2^n \operatorname{vol}(K).$$
This was proven in the following paper: convex bodies associated with a given convex body.
I'm looking for references, where a different/new proof of the above inequality is presented.
Edit. The motivation for this is the following: there are several analogues or finer version of such inequalities in Euclidean Geometry. I was wondering about a possibility of extending the ideas to a general Riemannian space, such as the Hyperbolic space---the original proof is not so helpful in creating a generalization. Here is an alternate link to the paper.