Journal of Convex Analysis, Vol. 4, No. 2, pp. 373-379 (1997)

Geometric Approximation of Proximal Normals

M. L. Radulescu and F. H. Clarke

Department of Mathematics, University of British Columbia, Vancouver BC V6T 1Z2, Canada,, and Centre de recherches math{é}matiques, Universit{é} de Montr{é}al, C. P.6128, succ. Centre-ville, Montr{é}al QC H3C 3J7, Canada,

Abstract: For $x\in H\setminus S$ and $\delta \ge 0$, the $\delta$-projection of $x$ onto $S$, is the set $\operatorname{proj}_S^\delta(x):=\left\{s\in S \colon \|s-x\|^2 \le d_S(x)^2 + \delta^2 \right\}.$ We prove that each vector $x-s$ with $s\in\operatorname{proj}_S^\delta(x)$ can be approximated by some nearby proximal normal. We also give a simple proof (new in the context of an infinite dimensional Hilbert space) of a result due to Rockafellar [17] concerning the approximation of "horizontal" normals to the epigraph of a lower semicontinuous function by "non-horizontal" ones.

Keywords: Nonsmooth analysis, distance function, $\delta$-projection, proximal normal, proximal subdifferential

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