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# Convex functions on Banach spaces not containing $\ell_1$

Published:1997-03-01
Printed: Mar 1997
• Jon Borwein
• Jon Vanderwerff
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## Abstract

There is a sizeable class of results precisely relating boundedness, convergence and differentiability properties of continuous convex functions on Banach spaces to whether or not the space contains an isomorphic copy of $\ell_1$. In this note, we provide constructions showing that the main such results do not extend to natural broader classes of functions.
 MSC Classifications: 46A55 - Convex sets in topological linear spaces; Choquet theory [See also 52A07] 46B20 - Geometry and structure of normed linear spaces 52A41 - Convex functions and convex programs [See also 26B25, 90C25]