Let $K \subset \mathbb{R}^2$ be a convex,compact with non empty interior
What is the regularity of $\partial K$ such that any supporting hyperplane of $K$ is the tangent?
this property is obvious if $K$ is a Disque, I guess that if $\partial K$ is $C^{2}$ we can obtain this property Thank you in advance for any answer, comments, suggestions or references.
the definition of the supporting hyperplane:
Let $K$ be a closed and bounded convex set in $\mathbb{R}^2$. Then a hyperplane $H$is called a supporting hyperplane of $K$ if and only if
(1) $K$is contained in one of the halfspaces of $H$, and
(2) $K\cap H \neq \emptyset $.