Convexity of a convex set minus points close to its frontier.

Question

Let $d\geq 1$. Let $K\subset\Bbb{R}^d$ be a convex set, $\varepsilon>0$, and $K(\varepsilon)\subset K$ the points of $K$ whose distance to $\partial K$ is less than $\varepsilon$.

Is $K\setminus K(\varepsilon)$ convex?

Answer 1

Let $A,B$ be points in $K\setminus K(\epsilon)$ and $C$ a convex combination of $A$ and $B$. Assume the distance of $C$ to $\partial K$ is less than epsilon, say we have $D\notin K$ with $d(C,D)<\epsilon$. Then $A':=A+(D-C)$ and $B':=B+(D-C)$ are still in $K$ and $D$ is a convex combination of $A'$ and $B'$, contradiction.