  Given a space X, we study the homotopy type of ${\mathcal B}_n(X)$ the space
obtained as "the union of all (n-1)-simplexes spanned by points in X". This is
a space encountered in non-linear analysis under the name of "space of
barycenters" or in differential geometry in the case n=2 as the "space of
chords". We first relate this space to a more familiar symmetric join
construction and then determine its stable homotopy type in terms of the
symmetric products on the suspension of X. This leads to a complete
understanding of the homology of ${\mathcal B}_n(X)$ as a functor of X, and to
an expression for its Euler characteristic given in terms of that of $X$. A
sharp connectivity theorem is also established. Finally barycenter spaces of
spheres are studied in details.
