The set of umbilics of infinite curvature of a "typical" convex body in $\mathbb{R}^n$
has measure zero, but is dense and uncountable.

Here "typical" is in the sense of Baire category: the subset of convex bodies
with this property is "comeager."

> Schneider, Rolf. "Curvatures of typical convex bodies—The complete picture." *Proc. Amer. Math. Soc.*, ([PDF download link][1]).

There are several properties of typical convex bodies that are somewhat counterintuitive.


  [1]: http://home.mathematik.uni-freiburg.de/rschnei/CurvaturePicture.pdf