A convex body with k-th mean curvature at least the sharp volume-to-surface constant at almost all normal points must be a ball, and constant curvature outside a small singular set also forces a ball.
Hessian measures of semi-convex functions and applications to support measures of convex bodies
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Uniqueness of singular convex hypersurfaces with lower bounded k-th mean curvature
A convex body with k-th mean curvature at least the sharp volume-to-surface constant at almost all normal points must be a ball, and constant curvature outside a small singular set also forces a ball.