Disks in Curves of Bounded Convex Curvature

TL;DR

This paper proves that any simple closed curve with bounded convex curvature in the plane always encloses an open unit disk within its interior, establishing a geometric property of such curves.

Contribution

It introduces the concept of bounded convex curvature for plane curves and proves that their interiors always contain an open unit disk, a new geometric insight.

Findings

Curves of bounded convex curvature always contain an open unit disk in their interior.

The paper formalizes the notion of bounded convex curvature for simple closed curves.

It establishes a geometric property linking curvature bounds to inscribed disks.

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 $\varepsilon_x>0$ 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.