Uniqueness of singular convex hypersurfaces with lower bounded k-th mean curvature

Abstract

We obtain a sharp characterization of the Euclidean ball among all convex bodies K whose boundary has a pointwise k-th mean curvature not smaller than a geometric constant at almost all normal points. This geometric constant depends only on the volume and the boundary area of K. We deduce this characterization from a new isoperimetric-type inequality for arbitrary convex bodies, for which the equality is achieved uniquely by balls. This second result is proved in a more general context of generalized mean-convex sets. Finally we positively answer a question left open in [FLW19] proving a further sharp characterization of the ball among all convex bodies that are of class $ \mathcal{C}^{1,1} outside a singular set, whose Hausdorff dimension is suitably bounded from above.

arXiv commentThe v4 version contains new results in theory of convex bodies. New title and new abstract. Submitted

References in corpus (2)

• Alexandrov's theorem revisited
• On representation of boundary integrals involving the mean curvature for mean-convex domains

Cited by in corpus (2)

• The double and triple bubble problem for stationary varifolds: the convex case
• Anisotropic curvature measures and uniqueness of convex bodies