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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 
$\text{PL}$-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 $\text{PL}$-convex for 
$1\,\le \,p\,\le \,2$.
