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AbstractIn this paper, we investigate an extremum problem for the power moment of a convex polygon contained in a disc. Our result is a generalization of a classical theorem: among all convex n-gons contained in a given disc, the regular n-gon inscribed in the circle (up to rotation) uniquely maximizes the area functional. It also implies that, among all convex n-gons contained in a given disc and containing the center in those interiors, the regular n-gon inscribed in the circle (up to rotation) uniquely maximizes the mean of the length of the chords passing through the center of the disc.
Advances in Geometry – de Gruyter
Published: Oct 26, 2021
Keywords: Extremum problem; power moment; convex polygon; dual cross-sectional measure; Jensen’s inequality; 52A40; 52A10; 51M16; 51M20
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