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On the sum of powered
distances to certain sets of points on the circle
Nikolai Nikolov and Rafael Rafailov |
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Vol. 253 (2011), No. 1, 157–168
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Abstract |
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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 P1…Pn. We use elementary analytic and purely geometric methods in the proof. |
Additional material |
Keywords
powered distance, unit circle
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Mathematical Subject Classification 2010
Primary: 52A40
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Milestones
Received: 27 December 2010
Revised: 16 July 2011
Accepted: 22 August 2011
Published: 28 November 2011
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Authors |
| Nikolai Nikolov | |
| Institute of Mathematics and
Informatics Bulgarian Academy of Sciences Acad. G. Bonchev 8, 1113 Sofia Bulgaria |
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| Rafael Rafailov | |
| Sofia High School of
Mathematics Iskar 61, 1000 Sofia Bulgaria |
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