Geodesic
Ranking small regular polygons by area and by perimeter
Diskretnyj analiz i issledovanie operacij, Tome 15 (2008) no. 3, pp. 65-73.

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From the pentagon onwards, for each odd number $n$ the area of the regular convex polygon with $n$ sides and unit diameter is greater than the area of the similar polygon with $n+1$ sides. Moreover, from the heptagon onwards, the difference in areas decreases when $n$ increases. Similar properties hold for the perimeter. A new proof of the Reinhardt's result is obtained. Tabl. 1, illustr. 1, bibl. 18.
Mots-clés : polygon
Keywords: diameter, area, perimeter. 
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Ch. Audet; P. Hansen; F. Messine. Ranking small regular polygons by area and by perimeter. Diskretnyj analiz i issledovanie operacij, Tome 15 (2008) no. 3, pp. 65-73. http://geodesic.mathdoc.fr/item/DA_2008_15_3_a6/

[1] Audet C., Hansen P., Jaumard B., Savard G., “A branch and cut algorithm for nonconvex quadratically constrained quadratic programming”, Mathematical programing. Ser. A, 87 (2000), 131–152 | MR | Zbl

[2] Audet C., Hansen P., Messine F., “The small octagon with longest perimeter”, J. Combin. Theory. Ser. A, 114 (2007), 135–150 | DOI | MR | Zbl

[3] Audet C., Hansen P., Messine F., “Extremal problems for convex polygons”, J. Global Optimization, 38 (2007), 163–179 | DOI | MR | Zbl

[4] Audet C., Hansen P., Messine F., Xiong J., “The largest small octagon”, J. Combinat. Theory. Ser. A, 98 (2002), 46–59 | DOI | MR | Zbl

[5] Blåsjö V., “The isoperimetric problem”, American Math. Monthly, 112 (2005), 526–566 | MR

[6] Brass P., Moser W., Pach J., Research problems in discrete geometry, Springer–Verl., New-York, 2005 | MR

[7] Bondarenko A., Bortz D. M., Moré J. J., A Collection of large-scale nonlinearly constrained optimization test problems, Argonne National Laboratory Research Report, August 20, 1998

[8] Conn A. R., Gould N. I. M., Toint P., LANCELOT, Springer Ser. Comput. Math., 17, Springer–Verl., New York, 1992 | MR | Zbl

[9] Croft H. T., Falconer K. J., Guy R. K., Unsolved problems in geometry, Springer–Verl., New York, 1991 | MR | Zbl

[10] Dolan E. D., Moré J. J., Benchmarking optimization software with COPS, Argonne National Laboratory Research Report, November 2000, revised January 2, 2001

[11] Dolan E. D., Moré J. J., Munson T. S., Benchmarking optimization software with COPS 3.0, Argonne National Laboratory Research Report, February, 2004

[12] Gill P. E., Murray W., Saunders M. A., User's guide for SNOPT 5.3: A Fortran package for large-scale nonlinear programming, report NA97-5, University of California, 1997

[13] Graham R. L., “The largest small hexagon”, J. Combinat. Theory. Ser. A, 18 (1975), 165–170 | DOI | MR | Zbl

[14] Messine F., Lagouanelle J. L., “Enclosure methods for multivariate differentiable functions and application to global optimization”, J. Universal Computer Science, 4 (1998), 589–603 | MR | Zbl

[15] Mossinghoff M. J., “A \$1 problem”, Amer. Math. Monthly, 113 (2006), 385–402 | MR

[16] Mossinghoff M. J., “Isodiametric problems for polygons”, Discrete and Comput. Geometry, 36 (2006), 363–379 | DOI | MR | Zbl

[17] Reinhardt K., “Extremale polygone gegebenen durchmessers”, Jahresber. Deutsch. Math. Verein., 31 (1922), 251–270

[18] Tamvakis N. K., “On the perimeter and the area of the convex polygon of a given diameter”, Bulletin of the Greek Mathematical Society, 28 (1987), 115–132 | MR | Zbl