Potential Theory of the Farthest-Point Distance Function

Richard S. Laugesen, Igor E. Pritsker

Research output: Contribution to journalArticle

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 σE has many interesting properties that reflect the topology and geometry of the compact set E. We prove σE(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 σE(E) = 1/2 for smooth convex sets of constant width. We conjecture σE(E) ≤ 1/2 for all E.

Original languageEnglish (US)
Pages (from-to)373-387
Number of pages15
JournalCanadian Mathematical Bulletin
Volume46
Issue number3
DOIs
StatePublished - 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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