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

Infinite-dimensional flats in the space of positive metrics on an ample line bundle