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28 Charles Audet, Pierre Hansen, and Frédéric Messine••••••••••• ••••Regular octagonΠ=8 sin( π 8 )≈3.0615...••Equal sidesΠ≈3.0956...•• •••Unequal sidesΠ≈3.1211...••Figure 5.Three small octagons[4] C. Audet, P. Hansen, F. Messine, and J. Xiong. The largest small octagon. J. Combin. Theory Ser. A, 98(1):46–59,2002.[5] A. Bezdek and F. Fodor. On convex polygons of maximal width. Arch. Math., 74(1):75–80, 2000.[6] B. Datta. A discrete isoperimetric problem. Geometriae Dedicata, 64: 55–68, 1997.[7] R. L. Graham. The largest small hexagon. J. Combinatorial Theory Ser. A, 18:165–170, 1975.[8] D. G. Larman and N. K. Tamvakis. The decomposition of the n-sphere and the boundaries of plane convexdomains. In Convexity and graph theory (Jerusalem, 1981), volume 87 of North-Holland Math. Stud., pages 209–214. North-Holland, Amsterdam, 1984.[9] F. Messine. Deterministic Global Optimization using Interval Constraint Propagation Techniques. RAIROOperations Research, Vol. 38, N. 4, 2004.[10] F. Messine. A Deterministic Global Optimization Algorithm for Design Problems. In C. Audet, P. Hansen, G.Savard (editors), Essays and Surveys in Global Optimization, Kluwer, pp. 267-294, 2005.[11] F. Messine., J.L. Lagouanelle. Enclosure Methods for Multivariate Differentiable Functions and Applicationto Global Optimization. Journal of Universal Computer Science, Vol. 4, (N.6) Springer-Verlag, pp. 589-603, 1998.[12] R.E. Moore. Interval Analysis. Prentice Hall, Englewood Cliffs, NJ, 1966.[13] H. Ratschek, J. Rokne. New Computer Methods for Global Optimization. Ellis Horwood, Chichester, 1988.[14] K. Reinhardt. Extremale polygone gegebenen durchmessers. Jahresber. Deutsch. Math. Verein, 31:251–270,1922.[15] N.K. Tamvakis. On the perimeter and the area of the convex polygon of a given diameter. Bull. Greek Math.Soc., 28: 115–132, 1987.[16] S. Vincze. On a geometrical extremum problem. Acta Sci. Math. Szeged, 12:136–142, 1950.[17] D.R. Woodall Thrackles and Deadlock. In Combinatorial Mathematics and Its Applications (D. J. A. Welsh, Ed.),Academic Press, New York, 1971.