Abstract
When dealing with convex functions defined on a normed vector space X the biconjugate is usually considered with respect to the dual system (X, X *), that is, as a function defined on the initial space X. However, it is of interest to consider also the biconjugate as a function defined on the bidual X **. It is the aim of this note to calculate the biconjugate of the functions obtained by several operations which preserve convexity. In particular we recover the result of Fitzpatrick and Simons on the biconjugate of the maximum of two convex functions with a much simpler proof.
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Zălinescu, C. On the second conjugate of several convex functions in general normed vector spaces. J Glob Optim 40, 475–487 (2008). https://doi.org/10.1007/s10898-007-9185-6
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DOI: https://doi.org/10.1007/s10898-007-9185-6
Keywords
- Biconjugate
- Composition with linear operators
- Conjugate
- Convex function
- Convolution
- Max-convolution
- Maximum of convex functions
- Normed vector space
- Sum of convex functions