In a finite-dimensional space with a smooth strictly convex norm, the distance to any convex body is differentiable outside the body, with norm gradient equal to the normalized outer normal of the metric projection.
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Convex analysis in normed spaces and metric projections onto convex bodies
In a finite-dimensional space with a smooth strictly convex norm, the distance to any convex body is differentiable outside the body, with norm gradient equal to the normalized outer normal of the metric projection.