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If $f$ is strongly convex and defined in a nbd of $\bar D$, it's gradient $\nabla f$ is a (monotone) diffeomorphism, so $\partial W$ is $C^2$ if $\partial D$ is $C^2$ and $f$ is $C^3$ on a nbd of $\bar D$. –  Pietro Majer Jun 2 '11 at 17:26
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