An extremum problem for the power moment of a convex polygon contained in a disc
-
Irmina Herburt
Abstract
In 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.
Funding statement: The second-named author is partially supported by JSPS Kakenhi (Grant Number 17K14191), JSPS Overseas Research Fellowships (Fellow Number 201860263) and the funds (Number 197102) from the Central Research Institute of Fukuoka University
Communicated by: R. Löwen
Acknowledgements
The authors would like to express their deep gratitude to the anonymous reviewer(s) for constructive suggestions on presentation.
References
[1] G. D. Chakerian, L. H. Lange, Geometric extremum problems. Math. Mag. 44 (1971), 57–69. MR301638 Zbl 0209.2650310.1080/0025570X.1971.11976102Search in Google Scholar
[2] L. Fejes Tóth, New proof of a minimum property of the regular n-gon. Amer. Math. Monthly 54 (1947), 589. MR24154 Zbl 0029.3110110.2307/2304499Search in Google Scholar
[3] L. Fejes Tóth, Regular figures, Pergamon Press, Macmillan Co., New York 1964. MR0165423 Zbl 0134.15705Search in Google Scholar
[4] A. Florian, Extremum problems for convex discs and polyhedra. In: Handbook of convex geometry, Vol. A, 177–221, North-Holland 1993. MR1242980 Zbl 0799.5200710.1016/B978-0-444-89596-7.50011-0Search in Google Scholar
[5] C. M. Fulton, S. K. Stein, Parallelograms inscribed in convex curves. Amer. Math. Monthly 67 (1960), 257–258. MR119148 Zbl 0090.1270310.2307/2309688Search in Google Scholar
[6] R. J. Gardner, Geometric tomography, volume 58 of Encyclopedia of Mathematics and its Applications. Cambridge Univ. Press 2006. MR2251886 Zbl 1102.52002Search in Google Scholar
[7] I. Herburt, M. Moszyńska, Z. Peradzyński, Remarks on radial centres of convex bodies. Math. Phys. Anal. Geom. 8 (2005), 157–172. MR2160333 Zbl 1089.5200110.1007/s11040-005-2968-4Search in Google Scholar
[8] A. G. Horváth, Z. Lángi, Maximum volume polytopes inscribed in the unit sphere. Monatsh. Math. 181 (2016), 341–354. MR3539939 Zbl 1354.5201610.1007/s00605-016-0949-2Search in Google Scholar
[9] M. S. Klamkin, D. J. Newman, The philosophy and applications of transform theory. SIAM Rev. 3 (1961), 10–36. MR119043 Zbl 0098.0040310.1137/1003002Search in Google Scholar
[10] J. Kürschák, Ueber dem Kreise ein- und umgeschriebene Vielecke. Math. Ann. 30 (1887), 578–581. MR1510466 JFM 19.0537.0210.1007/BF01444099Search in Google Scholar
[11] E. Lutwak, Mean dual and harmonic cross-sectional measures. Ann. Mat. Pura Appl. (4) 119 (1979), 139–148. MR551220 Zbl 0411.5200410.1007/BF02413172Search in Google Scholar
[12] J. O’Hara, Renormalization of potentials and generalized centers. Adv. in Appl. Math. 48 (2012), 365–392. MR2873883 Zbl 1255.3100710.1016/j.aam.2011.09.003Search in Google Scholar
[13] J. O’Hara, G. Solanes, Regularized Riesz energies of submanifolds. Math. Nachr. 291 (2018), 1356–1373. MR3817322 Zbl 1398.5301010.1002/mana.201600083Search in Google Scholar
[14] R. Schneider, Convex bodies: the Brunn–Minkowski theory, volume 151 of Encyclopedia of Mathematics and its Applications. Cambridge Univ. Press 2014. MR3155183 Zbl 1287.52001Search in Google Scholar
© 2021 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- An 𝔽p2-maximal Wiman sextic and its automorphisms
- Pseudo-algebraic Ricci solitons on Einstein nilradicals
- The wobbly divisors of the moduli space of rank-2 vector bundles
- Symmetries of complex analytic vector fields with an essential singularity on the Riemann sphere
- Betti numbers and pseudoeffective cones in 2-Fano varieties
- The generating rank of a polar Grassmannian
- The Beckman–Quarles theorem via the triangle inequality
- On Huisman’s conjectures about unramified real curves
- Geodesic orbit Finsler spaces with K ≥ 0 and the (FP) condition
- On the Segre invariant for rank two vector bundles on ℙ2
- Lifting coarse homotopies
- How to construct all metric f-K-contact manifolds
- An extremum problem for the power moment of a convex polygon contained in a disc
- Sharply transitive sets in PGL2(K)
Articles in the same Issue
- Frontmatter
- An 𝔽p2-maximal Wiman sextic and its automorphisms
- Pseudo-algebraic Ricci solitons on Einstein nilradicals
- The wobbly divisors of the moduli space of rank-2 vector bundles
- Symmetries of complex analytic vector fields with an essential singularity on the Riemann sphere
- Betti numbers and pseudoeffective cones in 2-Fano varieties
- The generating rank of a polar Grassmannian
- The Beckman–Quarles theorem via the triangle inequality
- On Huisman’s conjectures about unramified real curves
- Geodesic orbit Finsler spaces with K ≥ 0 and the (FP) condition
- On the Segre invariant for rank two vector bundles on ℙ2
- Lifting coarse homotopies
- How to construct all metric f-K-contact manifolds
- An extremum problem for the power moment of a convex polygon contained in a disc
- Sharply transitive sets in PGL2(K)
