Involve: A Journal of Mathematics

  • Involve
  • Volume 5, Number 3 (2012), 327-338.

The $n$-diameter of planar sets of constant width

Zair Ibragimov and Tuan Le

Full-text: Open access

Abstract

We study the notion of n-diameter for sets of constant width. A convex set in the plane is said to be of constant width if the distance between two parallel support lines is constant, independent of the direction. The Reuleaux triangles are the well-known examples of sets of constant width that are not disks. The n-diameter of a compact set E in the plane is

d n ( E ) = max 1 i < j n | z i z j | 2 n ( n 1 ) ,

where the maximum is taken over all zkE, k=1,2,,n. We prove that if n=5, then the Reuleaux n-gons have the largest n-diameter among all sets of given constant width. The proof is based on the solution of an extremal problem for n-diameter.

Article information

Source
Involve, Volume 5, Number 3 (2012), 327-338.

Dates
Received: 12 September 2011
Revised: 14 December 2011
Accepted: 16 December 2011
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.involve/1513733501

Digital Object Identifier
doi:10.2140/involve.2012.5.327

Mathematical Reviews number (MathSciNet)
MR3044618

Zentralblatt MATH identifier
1277.30014

Subjects
Primary: 30C65: Quasiconformal mappings in $R^n$ , other generalizations
Secondary: 05C25: Graphs and abstract algebra (groups, rings, fields, etc.) [See also 20F65]

Keywords
$n$-diameter constant width sets Pólya extremal problem

Citation

Ibragimov, Zair; Le, Tuan. The $n$-diameter of planar sets of constant width. Involve 5 (2012), no. 3, 327--338. doi:10.2140/involve.2012.5.327. https://projecteuclid.org/euclid.involve/1513733501


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