Let $d\geq 1$. Let $K\subset\Bbb{R}^d$ be a convex compact set with smooth boundary. Let $\varepsilon>0$, and let $K(\varepsilon)$ denote set set of points of $K$ whose distance to $\partial K$ is less than $\varepsilon$.
Let $K_0=K\setminus K(\varepsilon)$. I have to show that $K_0$ is a convex compact set with smooth boundary. I already showed it's compact and convex, but I don't know how to prove its boudary is smooth.