Stability of slopes and subdifferentials (Q1405597)

Given a Banach space \(X\) and a function \(f:X\rightarrow\mathbb{R\cup \{+\infty\}}\), its slope is the function defined by \(\text{slope} f(x)=\lim\sup_{y\rightarrow x,y\neq x}\frac{(f(x)-f(y))^{+}}{\left\| x-y\right\| }\) where \(\alpha^{+}=\max\{0,\alpha\}\) for \(x\in \text{dom}f\), while \(\text{slope}f(x)=+\infty\) for \(x\notin\text{dom}f\). In this paper, stability of the slope of lsc functions under the so-called ba-convergence is studied (ba stands for ball-affine). This convergence is stronger than epi-convergence, but weaker than the Attouch-Wets convergence, the slice convergence and other notions of variational convergence. Also, for convex functions a kind of converse is shown, of the form: if the slope is upper epi-stable under a perturbation of a convex lsc function in the space of convex lsc functions, then the perturbation must be ball-affine.