C ² -Lusin Approximation of Strongly Convex Bodies

We prove that, if

is a locally strongly convex body (not necessarily compact), then for any open set

V \supset \partial W V ⊃ ∂ W

and

\varepsilon>0 ε > 0

, there exists a

C^2 C 2

locally strongly convex body

W_{\varepsilon, V} W ε , V

such that

\mathcal{H}^{n-1}(\partial W_{\varepsilon, V}\triangle\,\partial W)<\varepsilon H n − 1 ( ∂ W ε , V △   ∂ W ) < ε

and

\partial W_{\varepsilon, V}\subset V ∂ W ε , V ⊂ V

. Moreover, if

W W

is strongly convex, then

W_{\varepsilon, V} W ε , V

is strongly convex as well.