## Weak* Hypertopologies with Application to Genericity of Convex Sets

##### Abstract

We propose a new class of hypertopologies, called here weak$^{\ast }$
hypertopologies, on the dual space $\mathcal{X}^{\ast }$ of a real or
complex topological vector space $\mathcal{X}$. The most well-studied and
well-known hypertopology is the one associated with the Hausdorff metric for
closed sets in a complete metric space. Therefore, we study in detail its
corresponding weak$^{\ast }$ hypertopology, constructed from the Hausdorff
distance on the field (i.e. $\mathbb{R}$ or $\mathbb{C}$) of the vector
space $\mathcal{X}$ and named here the weak$^{\ast }$-Hausdorff
hypertopology. It has not been considered so far and we show that it can
have very interesting mathematical connections with other mathematical
fields, in particular with mathematical logics. We explicitly demonstrate
that weak$^{\ast }$ hypertopologies are very useful and natural structures\
by using again the weak$^{\ast }$-Hausdorff hypertopology in order to study
generic convex weak$^{\ast }$-compact sets in great generality. We show that
convex weak$^{\ast }$-compact sets have generically weak$^{\ast }$-dense set
of extreme points in infinite dimensions. An extension of the well-known
Straszewicz theorem to Gateaux-differentiability (non necessarily Banach)
spaces is also proven in the scope of this application.