  We show that any continuous positive metric on an ample line bundle L lies at
the apex of many infinite-dimensional Mabuchi-flat cones. More precisely, given
any bounded graded filtration F of the section ring of L, the set of bounded
decreasing convex functions on the support of the Duistermaat--Heckman measure
of F embeds L^p-isometrically into the space of bounded positive metrics on L
with respect to Darvas' d_p distance for all p\in[1,\infty), and in particular
with respect to the Mabuchi metric (p=2).
