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Regularisations of convex functions and slicewise suprema

Published online by Cambridge University Press:  17 April 2009

S. Simons
S. Simons
Affiliation:
Department of MathematicsUniversity of CaliforniaSanta Barbara CA 93106–3080United States of America
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Abstract

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For a number of years, there has been interest in the regularisation of a given proper convex lower semicontinuous function on a Banach space, defined to be the episum (=inf-convolution) of the function with a scalar multiple of the norm. There is an obvious geometric way of characterising this regularisation as the lower envelope of cones lying above the graph of the original function. In this paper, we consider the more interesting problem of characterising the regularisation in terms of approximations from below, expressing the regularisation as the upper envelope of certain subtangents to the graph of the original function. We shall show that such an approximation is sometimes (but not always) valid. Further, we shall give an extension of the whole procedure in which the scalar multiple of the norm is replaced by a more general sublinear functional. As a by-product of our analysis, we are led to the consideration of two senses stronger than the pointwise sense in which a function on a Banach space can be expressed as the upper envelope of a family of functions. These new senses of suprema lead to some questions in Banach space theorey.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 50 , Issue 3 , December 1994 , pp. 481 - 499
Copyright
Copyright © Australian Mathematical Society 1994

References

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