  In previous work J. Backhoff-Veraguas, M. Beiglb\"ock and the present authors
showed that the notions of stretched Brownian motion and Bass martingale
between two probability measures on Euclidean space coincide if and only if
these two measures satisfy an irreducibility condition.
  Now we consider the general case, i.e. a pair of measures which are not
necessarily irreducible. We show that there is a paving of Euclidean space into
relatively open convex sets such that the stretched Brownian motion decomposes
into a (possibly uncountable) family of Bass martingales on these sets. This
paving coincides with the irreducible convex paving studied in previous work of
H. De March and N. Touzi.
