C²-Lusin approximation of strongly convex bodies

We prove that, if W ⊂ Rⁿ is a locally strongly convex body (not necessarily compact), then for any open set V ⊃ ∂ W and ε>0, and V ⊃ ∂ W is open, then there exists a C² locally strongly convex body Wε, V such that Hⁿ⁻¹(∂ Wε, V\,∂ W)<ε and ∂ Wε, V⊂ V. Moreover, if W is strongly convex, then Wε, V is strongly convex as well.