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

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

$n$-diameter constant width sets Pólya extremal problem


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.

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