Geometric Approximation of Proximal Normals

For x ∈ H ∖ S and δ ≥ 0, the δ-projection of x onto S, is the set $\mathrm{proj}^{\delta}_{S}(x) := \{s \in S : \|s - x\|^2 \leq d_S(x)^2 + \delta^2\}$. We prove that each vector x − s with $s \in \mathrm{proj}^{\delta}_{S}(x)$ 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.