A CRANDALL-PAZY EXTENSION OF THE SLOPE ATTOUCH THEOREM

Overview & Synopsis

For a proper lower semicontinuous convex function \(f\) on a real Hilbert space \(H\), let $$s_f(x)=\operatorname{dist}(0,\partial f(x)) \qquad\text{and}\qquad p_f=P_{\overline{\operatorname{ran}\partial f}}(0),$$ where \(P_C\) denotes the metric projection onto a nonempty closed convex set \(C\). Recent slope versions of Attouch theory show, when the limit function is bounded from below, that slope epi-convergence is equivalent to Painlevé–Kuratowski convergence of subdifferential graphs and, after normalization, characterizes convergence of the functions. We show that in these convergence results lower boundedness can be replaced by convergence of the Crandall–Pazy directions. In finite dimensions, epi-convergence of \(s_{f_n}\) to \(s_f\), together with \(p_{f_n}\to p_f\), implies graphical convergence of the subdifferentials; under the weak normalization used in the slope Attouch theorem, it also implies epi-convergence of \(f_n\) to \(f\). Under the compactness-type assumption of Pérez-Aros, Thibault and Zagrodny, the corresponding graph/slope equivalence extends to separable Hilbert spaces, and Mosco convergence is equivalent to slope epi-convergence together with weak normalization, provided \(p_{f_n}\to p_f\). The proof isolates the role of lower boundedness as a final identification step, which is replaced by Crandall–Pazy rigidity. Finally, slope epi-convergence gives $$\limsup_{n\to\infty}\|p_{f_n}\|\le \|p_f\|,$$ so the Crandall–Pazy compatibility is automatic when \(p_f=0\).