1. CMB 2011 (vol 55 pp. 697)
|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
2. CMB 2008 (vol 51 pp. 205)
|On GÃ¢teaux Differentiability of Pointwise Lipschitz Mappings |
We prove that for every function $f\from X\to Y$, where $X$ is a separable Banach space and $Y$ is a Banach space with RNP, there exists a set $A\in\tilde\mcA$ such that $f$ is G\^ateaux differentiable at all $x\in S(f)\setminus A$, where $S(f)$ is the set of points where $f$ is pointwise-Lipschitz. This improves a result of Bongiorno. As a corollary, we obtain that every $K$-monotone function on a separable Banach space is Hadamard differentiable outside of a set belonging to $\tilde\mcC$; this improves a result due to Borwein and Wang. Another corollary is that if $X$ is Asplund, $f\from X\to\R$ cone monotone, $g\from X\to\R$ continuous convex, then there exists a point in $X$, where $f$ is Hadamard differentiable and $g$ is Fr\'echet differentiable.
Keywords:GÃ¢teaux differentiable function, Radon-NikodÃ½m property, differentiability of Lipschitz functions, pointwise-Lipschitz functions, cone mononotone functions