Abstract or Additional Information
Convex functions pervade analysis. For example, any norm is a convex function. A real-valued convex function defined on an interval has several nice properties. Such functions are continuous. Such functions are differentiable almost everywhere. What if we consider convex functions defined on convex subsets of bigger Banach spaces? In many spaces, such functions will necessarily be differentiable on a dense $G_\delta$ set. We'll discuss some basics and see that geometric properties of the space, including the Radon-Nikodym property, get involved in guaranteeing this differentiability.