The right-invariant Riemannian metric on simplex shape spaces in fact makes them particular Riemannian
symmetric spaces of non-compact type. In the paper, the general properties of such symmetric spaces
are made explicit for simplex shape spaces. In particular, a global matrix coordinate representation is
suggested, with respect to which several geometric features, important for shape analysis, have simple
and easily computable expressions. As a typical application, it is shown how to locate the Fréchet means
of a class of probability measures on the simplex shape spaces, a result analogous to that for Kendall's
shape spaces.