A combinatorial proof of an identity involving Eulerian numbers

We give a combinatorial proof of an identity that involves Eulerian numbers and was obtained algebraically by Brenti and Welker (2009). To do so, we study alcoved triangulations of dilated hypersimplices. As a byproduct, we describe the dual graph of the triangulation in the case of the standard simplex, conjecture its structure for general hypersimplices, and prove combinatorially that the Eulerian numbers coincide with the normalized volumes of the hypersimplices.