Generalized roundness of the Schatten class, CpCp

In the paper Generalized roundness and negative type  , Lennard, Tonge, and Weston show that the geometric notion of generalized roundness in a metric space is equivalent to that of negative type. Using this equivalent characterization, along with classical embedding theorems, the authors prove that for p>2p>2, LpLp fails to have generalized roundness q   for any q>0q>0. It is noted, as a consequence, that the Schatten class CpCp, for p>2p>2, has maximal generalized roundness 0. In this paper, we prove that this result remains true for p   in the interval (0,2)(0,2).