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

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M. L. Radulescu and F. H. Clarke

Department of Mathematics, University of British Columbia, Vancouver BC V6T 1Z2, Canada,mihaela@math.ubc.ca, 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, clarke@crm.umontreal.ca

**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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