Disks in Curves of Bounded Convex Curvature

Abstract

We say that a simple, closed curve $\gamma$ in the plane has bounded convex curvature if for every point $x$ on $\gamma$, there is an open unit disk $U_x$ and ${\epsilon }_x>0$ such that $x\in \partial U_x$ and $B_{{\epsilon }_x}(x)\cap U_x\subset \text{Int}\gamma$. We prove that the interior of every curve of bounded convex curvature contains an open unit disk.

Accepted for publication in The American Mathematical Monthly