The norm on a Banach space gives rise to a subharmonic function on the complex plane for which the distributional Laplacian gives a Riesz measure. This measure is calculated explicitly here for Lebesgue L^p spaces and the von Neumann-Schatten trace ideals. Banach spaces that are q -uniformly PL -convex in the sense of Davis, Garling and Tomczak-Jaegermann are characterized in terms of the mass distribution of this measure. This gives a new proof that the trace ideals c^p are 2-uniformly PL -convex for 1≤p≤2 .