### Abstract

We study the farthest-point distance function, which measures the distance from z ε ℂ to the farthest point or points of a given compact set E in the plane. The logarithm of this distance is subharmonic as a function of z, and equals the logarithmic potential of a unique probability measure with unbounded support. This measure σ_{E} has many interesting properties that reflect the topology and geometry of the compact set E. We prove σ_{E}(E) ≤ 1/2 for polygons inscribed in a circle, with equality if and only if E is a regular n-gon for some odd n. Also we show σ_{E}(E) = 1/2 for smooth convex sets of constant width. We conjecture σ_{E}(E) ≤ 1/2 for all E.

Original language | English (US) |
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Pages (from-to) | 373-387 |

Number of pages | 15 |

Journal | Canadian Mathematical Bulletin |

Volume | 46 |

Issue number | 3 |

DOIs | |

State | Published - Sep 2003 |

### Keywords

- Convex bodies of constant width
- Distance function
- Farthest points
- Representing measure
- Subharmonic function

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

*Canadian Mathematical Bulletin*,

*46*(3), 373-387. https://doi.org/10.4153/CMB-2003-039-0