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Two moons in a puddle

Berk Ceylan

Abstract

We prove that a simple closed plane curve with curvature at most 1 and diameter at least 4, surrounds two disjoint open unit disks. We also pose an open question relating the diameter lower bound to the length of the curve.

Two moons in a puddle

Abstract

We prove that a simple closed plane curve with curvature at most 1 and diameter at least 4, surrounds two disjoint open unit disks. We also pose an open question relating the diameter lower bound to the length of the curve.
Paper Structure (2 theorems, 1 equation, 5 figures)

This paper contains 2 theorems, 1 equation, 5 figures.

Key Result

Lemma 1

Let $\gamma$ be a simple closed plane curve and $p$ be a point on it. Assume that $\gamma_1$ is an arc of $\gamma$ with only the end points touching $C_p$. Then there is a point $q$ of $\gamma_1$ such that the osculating circle at $q$ supports $\gamma$ from inside. Moreover, any supporting osculatin

Figures (5)

  • Figure 1: A curve that can only contain two disjoint unit circles.
  • Figure 2: Sketch of the lemma.
  • Figure 3: The idea is to apply lemma to $\gamma_1$ and its analogous arc for $C_{p_2}$.
  • Figure 4: Diameter is 4 and length is $2\pi+4$.
  • Figure 5: Three almost tangent unit circles.

Theorems & Definitions (4)

  • Lemma 1
  • proof
  • Theorem 1
  • proof