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Search: MSC category 49J50 ( Frechet and Gateaux differentiability [See also 46G05, 58C20] )

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1. CMB 2011 (vol 56 pp. 272)

Cheng, Lixin; Luo, Zhenghua; Zhou, Yu
On Super Weakly Compact Convex Sets and Representation of the Dual of the Normed Semigroup They Generate
In this note, we first give a characterization of super weakly compact convex sets of a Banach space $X$: a closed bounded convex set $K\subset X$ is super weakly compact if and only if there exists a $w^*$ lower semicontinuous seminorm $p$ with $p\geq\sigma_K\equiv\sup_{x\in K}\langle\,\cdot\,,x\rangle$ such that $p^2$ is uniformly Fréchet differentiable on each bounded set of $X^*$. Then we present a representation theorem for the dual of the semigroup $\textrm{swcc}(X)$ consisting of all the nonempty super weakly compact convex sets of the space $X$.

Keywords:super weakly compact set, dual of normed semigroup, uniform Fréchet differentiability, representation
Categories:20M30, 46B10, 46B20, 46E15, 46J10, 49J50

2. CMB 2011 (vol 55 pp. 697)

Borwein, Jonathan M.; Vanderwerff, Jon
Constructions of Uniformly Convex Functions
We give precise conditions under which the composition of a norm with a convex function yields a uniformly convex function on a Banach space. Various applications are given to functions of power type. The results are dualized to study uniform smoothness and several examples are provided.

Keywords:convex function, uniformly convex function, uniformly smooth function, power type, Fenchel conjugate, composition, norm
Categories:52A41, 46G05, 46N10, 49J50, 90C25

3. CMB 2000 (vol 43 pp. 25)

Bounkhel, M.; Thibault, L.
Subdifferential Regularity of Directionally Lipschitzian Functions
Formulas for the Clarke subdifferential are always expressed in the form of inclusion. The equality form in these formulas generally requires the functions to be directionally regular. This paper studies the directional regularity of the general class of extended-real-valued functions that are directionally Lipschitzian. Connections with the concept of subdifferential regularity are also established.

Keywords:subdifferential regularity, directional regularity, directionally Lipschitzian functions
Categories:49J52, 58C20, 49J50, 90C26

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