  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.
