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# Complex Uniform Convexity and Riesz Measure

Published:2004-04-01
Printed: Apr 2004
• Gordon Blower
• Thomas Ransford
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## Abstract

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 $1\leq p\leq 2$.
 Keywords: subharmonic functions, Banach spaces, Schatten trace ideals
 MSC Classifications: 46B20 - Geometry and structure of normed linear spaces 46L52 - Noncommutative function spaces