  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 $\varepsilon_x>0$ such that $x\in\partial U_x$ and
$B_{\varepsilon_x}(x)\cap U_x\subset\text{Int}\;\gamma$. We prove that the
interior of every curve of bounded convex curvature contains an open unit disk.
