### Abstract

Let P be a simple polygon. Let the vertices of P be mapped, according to a counterclockwise traversal of the boundary, into a strictly increasing sequence of real numbers in [0, 2π). Let a ray be drawn from each vertex so that the angle formed by the ray and a horizontal line pointing to the right equals, in measure, the number mapped to the vertex. Whenever the rays from two consecutive vertices intersect, let them induce the triangular region with extreme points comprising the vertices and the intersection point. It is shown that there is a fixed α such that if all of the assigned angles are increased by α the triangular regions induced by the redirected rays cover the interior of P. This covering implies the standard isoperimetric inequalities in two dimensions, as well as several new inequalities, and resolves a question posed by Yaglom and Boltanskiǐ.

Original language | English (US) |
---|---|

Pages (from-to) | 239-255 |

Number of pages | 17 |

Journal | Discrete and Computational Geometry |

Volume | 29 |

Issue number | 2 |

DOIs | |

State | Published - 2003 |

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### ASJC Scopus subject areas

- Theoretical Computer Science
- Computational Theory and Mathematics
- Discrete Mathematics and Combinatorics
- Geometry and Topology

### Cite this

**An isoperimetric theorem in plane geometry.** / Siegel, Alan.

Research output: Contribution to journal › Article

*Discrete and Computational Geometry*, vol. 29, no. 2, pp. 239-255. https://doi.org/10.1007/s00454-002-2809-1

}

TY - JOUR

T1 - An isoperimetric theorem in plane geometry

AU - Siegel, Alan

PY - 2003

Y1 - 2003

N2 - Let P be a simple polygon. Let the vertices of P be mapped, according to a counterclockwise traversal of the boundary, into a strictly increasing sequence of real numbers in [0, 2π). Let a ray be drawn from each vertex so that the angle formed by the ray and a horizontal line pointing to the right equals, in measure, the number mapped to the vertex. Whenever the rays from two consecutive vertices intersect, let them induce the triangular region with extreme points comprising the vertices and the intersection point. It is shown that there is a fixed α such that if all of the assigned angles are increased by α the triangular regions induced by the redirected rays cover the interior of P. This covering implies the standard isoperimetric inequalities in two dimensions, as well as several new inequalities, and resolves a question posed by Yaglom and Boltanskiǐ.

AB - Let P be a simple polygon. Let the vertices of P be mapped, according to a counterclockwise traversal of the boundary, into a strictly increasing sequence of real numbers in [0, 2π). Let a ray be drawn from each vertex so that the angle formed by the ray and a horizontal line pointing to the right equals, in measure, the number mapped to the vertex. Whenever the rays from two consecutive vertices intersect, let them induce the triangular region with extreme points comprising the vertices and the intersection point. It is shown that there is a fixed α such that if all of the assigned angles are increased by α the triangular regions induced by the redirected rays cover the interior of P. This covering implies the standard isoperimetric inequalities in two dimensions, as well as several new inequalities, and resolves a question posed by Yaglom and Boltanskiǐ.

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UR - http://www.scopus.com/inward/citedby.url?scp=0037661025&partnerID=8YFLogxK

U2 - 10.1007/s00454-002-2809-1

DO - 10.1007/s00454-002-2809-1

M3 - Article

AN - SCOPUS:0037661025

VL - 29

SP - 239

EP - 255

JO - Discrete and Computational Geometry

JF - Discrete and Computational Geometry

SN - 0179-5376

IS - 2

ER -