Vol. 253, No. 1, 2011

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On the sum of powered distances to certain sets of points on the circle

Nikolai Nikolov and Rafael Rafailov

Vol. 253 (2011), No. 1, 157–168

We consider an extremal problem in geometry. Let λ be a real number and let A, B and C be arbitrary points on the unit circle Γ. We give a full characterization of the extremal behavior of the function f(M,λ) = MAλ + MBλ + MCλ, where M is a point on the unit circle as well. We also investigate the extremal behavior of i=1nXPi, where the Pi, for i = 1,,n, are the vertices of a regular n-gon and X is a point on Γ, concentric to the circle circumscribed around P1Pn. We use elementary analytic and purely geometric methods in the proof.

Additional material
powered distance, unit circle
Mathematical Subject Classification 2010
Primary: 52A40
Received: 27 December 2010
Revised: 16 July 2011
Accepted: 22 August 2011
Published: 28 November 2011
Nikolai Nikolov
Institute of Mathematics and Informatics
Bulgarian Academy of Sciences
Acad. G. Bonchev 8, 1113 Sofia
Rafael Rafailov
Sofia High School of Mathematics
Iskar 61, 1000 Sofia