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A geometric property of quadrilaterals

Efstathios Konstantinos Chrontsios Garitsis, Aimo Hinkkanen

Abstract

Quadrilaterals in the complex plane play a significant part in the theory of planar quasiconformal mappings. Motivated by the geometric definition of quasiconformality, we prove that every quadrilateral with modulus in an interval $[1/K, K]$, where $K>1$, contains a disk lying in its interior, of radius depending only on the internal distances between the pairs of opposite sides of the quadrilateral and on $K$.

A geometric property of quadrilaterals

Abstract

Quadrilaterals in the complex plane play a significant part in the theory of planar quasiconformal mappings. Motivated by the geometric definition of quasiconformality, we prove that every quadrilateral with modulus in an interval , where , contains a disk lying in its interior, of radius depending only on the internal distances between the pairs of opposite sides of the quadrilateral and on .
Paper Structure (9 sections, 9 theorems, 81 equations, 11 figures)

This paper contains 9 sections, 9 theorems, 81 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Acknowledgements.
  3. Notation
  4. Preliminary reductions
  5. Avoiding the vertices of $Q$
  6. A consequence of Rengel's inequality
  7. Approximation of quadrilaterals
  8. Proof of Theorem \ref{['thm_sa']}
  9. Final remarks

Key Result

Theorem 1

For every $K\geq 1$ there is a constant $\delta\in (0,1)$ depending only on $K$ such that every quadrilateral $Q$ with $M(Q)\in [1/K, K]$ contains a disk of radius $\delta \max \{ s_a(Q), s_b(Q) \}$.

Figures (11)

  • Figure 1: An example of a set $K$ constructed inside a quadrilateral $Q$.
  • Figure 2: An illustration of what would happen if we assumed $\partial Q_\tau$ is not a Jordan curve.
  • Figure 3: Boundary points of $Q$ can only lie in one component and the shaded areas.
  • Figure 4: An example of $N_{C_\epsilon}$.
  • Figure 5: Showing how thin $N_{C_\epsilon}$ is chosen to be, even inside $D(w_0, R)$ and compared to $|z_+-w_+|$, $|z_- - w_-|$.
  • ...and 6 more figures

Theorems & Definitions (17)

  • Theorem 1
  • Lemma 1
  • proof
  • Proposition A
  • Remark 1
  • Lemma 2
  • proof
  • Remark 2
  • Theorem 2
  • Theorem 3
  • ...and 7 more