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Convex spaces: Classification by differentiable convex functions

Published online by Cambridge University Press:  17 April 2009

Roger Eyland  and
Bernice Sharp
Roger Eyland
Affiliation:
School of Maths and Statistics University of Sydney, New South Wales 2006, Australia
Bernice Sharp
Affiliation:
Australian Catholic University, 179 Albert Road Strathfield NSW 2135, Australia
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Abstract

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The differentiability, of a specified strength, of a convex function at a point, is shown to be characterised by the convergence of subdifferentials in the appropriate topology on the dual space. This is used to prove that if each gauge is densely differentiable then so is each convex function. The generic version of this is equivalent to a conjecture which, for Gateaux differentiability and Banach spaces, is the long standing open question of whether X × ℝ is Weak Asplund whenever X is. Some progress is made towards a resolution.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 46 , Issue 1 , August 1992 , pp. 127 - 138
Copyright
Copyright © Australian Mathematical Society 1992

References

[1]Asplund, E., ‘Frédict differentiability of convex functions’, Acta Math 121 (1968), 31–47.CrossRefGoogle Scholar
[2]Borwein, J.M., ‘Continuity and differentiability properties of convex operators’, Proc. London Math. Soc. 44 (1982), 420–444.CrossRefGoogle Scholar
[3]Čoban, M. and Kenderov, P., ‘Dense Gateaux differentiability of the sup-norm in C(T) and the topological properties of T’, C.R. Acad. Bulgare Sci. 38 (1985), 1603–1604.Google Scholar
[4]Eyland, R.W. and Sharp, B., ‘A factor theorem for locally convex differentiability spaces’, Bull. Austral. Math. Soc. 43 (1991), 101–113.CrossRefGoogle Scholar
[5]Giles, J.R., ‘Convex analysis with application in differentiation of convex functions’, Pitman Res. Notes Math. Ser. 58 (1982).Google Scholar
[6]Köthe, G., Topological vector spaces I (Springer-Verlag, Berlin, Heidelberg, New York, 1969).Google Scholar
[7]Larman, D.G. and Phelps, R., ‘Gateaux differentiability of convex functions on Banach spaces’, J. London Math. Soc. 20 (1979), 115–127.CrossRefGoogle Scholar
[8]Namioka, I. and Phelps, R., ‘Banach spaces which are Asplund spaces’, Duke Math. J. 42 (1975), 735–750.CrossRefGoogle Scholar
[9]Phelps, R., ‘Convex functions, monotone operators and differentiability’, in Lecture Notes in Math. 1364 (Springer-Verlag, Berlin, Heidelberg, New York, 1989).Google Scholar
[10]Phelps, R., Differentiability of convex functions on Banach spaces, Lecture Notes (University College, London, 1978).Google Scholar