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Rotundity, smoothness and duality

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Penot J.-P.

Control and Cybernetics

2003

tom Vol. 32, no 4

711-733

The duality between smoothness and rotundity of functions is studied in a nonlinear abstract framework. Here smoothness is enlarged to subdifferentiability properties and rotundity is formulated by means of approximation properties.

A duality for starshaped functions

100%

Penot J.-P.

2002

tom Vol. 50, no 2

127-139

A conjugacy is introduced for the class of starshaped functions from [0, infinity] into [0, infinity], i.e. the class of functions f such that their slope s : t --> f (t)/t is nondecreasing. This class is stable by several operations and plays a key role in the study of uniformly convex and uniformly smooth convex functions and in the geometry of Banach spaces. Here the inversion of the subdifferential as in the Legendre-Fenchel transform is replaced by an inversion device of the slope s which uses the ordering of R.