The space of convex combinations of things.
Hacks for it.
This one is apparently “folk wisdom”.
But say you wish to simulate a vector drawn uniformly from the
\(n\)simplex.

simulate \(n\) random uniform variables on the unit interval,
\((u_1,_u_2,\dots,u_n)\)

Sort them in decreasing order, \((u'_1,_u'_2,\dots,u'_n)\)

Your random vector is \((u'_10, u'_2u'_1, u'_3u'_2,\dots,u'_nu'_{n1})\)
This surprisingly relates to Dirichlet variables. Can you see how?
Refs