DOI: 10.68381/jca04022 ISSN: 0944-6532
Geometric Approximation of Proximal Normals
M. L. Radulescu, F. H. Clarke
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\}
p
r
o
j
S
δ
(
x
)
:
=
{
s
∈
S
:
∥
s
−
x
∥
2
≤
d
S
(
x
)
2
+
δ
2
}
. We prove that each vector x − s with
s \in \mathrm{proj}^{\delta}_{S}(x)
s
∈
p
r
o
j
S
δ
(
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.