# American Institute of Mathematical Sciences

July  2016, 21(5): 1587-1601. doi: 10.3934/dcdsb.2016012

## A generalization of the Blaschke-Lebesgue problem to a kind of convex domains

 1 Department of Mathematics, Tongji University, Shanghai 200092, China 2 School of Mathematical Sciences, Huaibei Normal University, Huaibei 235000, Anhui Province, China 3 Chengdu No.7 High School, Chengdu 610041, Sichuan Province, China

Received  October 2013 Revised  March 2014 Published  April 2016

In this paper we will introduce for a convex domain $K$ in the Euclidean plane a function $\Omega_{n}(K, \theta)$ which is called by us the biwidth of $K$, and then try to find out the least area convex domain with constant biwidth $\Lambda$ among all convex domains with the same constant biwidth. When $n$ is an odd integer, it is proved that our problem is just that of Blaschke-Lebesgue, and when $n$ is an even number, we give a lower bound of the area of such constant biwidth domains.
Citation: Shengliang Pan, Deyan Zhang, Zhongjun Chao. A generalization of the Blaschke-Lebesgue problem to a kind of convex domains. Discrete and Continuous Dynamical Systems - B, 2016, 21 (5) : 1587-1601. doi: 10.3934/dcdsb.2016012
##### References:
 [1] H. Anciaux and N. Georgiou, The Blaschke-Lebesgue problem for constant width bodies of revolution, 2009. preprint, arXiv:0903.4284. [2] H. Anciaux and B. Guilfoyle, On the three-dimensional Blaschke-Lebesgue problem, Proc. Amer. Math. Soc., 139 (2011), 1831-1839. doi: 10.1090/S0002-9939-2010-10588-9. [3] T. Bayen, Analytical parameterization of rotors and proof of a Goldberg conjecture by optimal control theory, SIAM J. Control Optim., 47 (2008), 3007-3036. doi: 10.1137/070705325. [4] T. Bayen, T. Lachand-Robert and E. Oudet, Analytic parametrizations and volume minimization of three dimensional bodies of constant width, Arch. Ration. Mech. Anal., 186 (2007), 225-249. doi: 10.1007/s00205-007-0060-x. [5] W. Blaschke, Konvexe Bereiche gegebener konstanter Breite und kleinsten, Inhalts, Math. Ann., 76 (1915), 504-513. doi: 10.1007/BF01458221. [6] W. Blaschke, Kreis und Kugel, $2^{nd}$ edition, Gruyter, Berlin, 1956. [7] S. Campi, A. Colesanti and P. Gronchi, Minimum problems for volumes of convex bodies, in Partial Diferential Equations and Applications (eds. P. Marcellini, G. Talenti, and E. Visintin), Marcel-Dekker, New York, 177 (1996), 43-55. [8] G. D. Chakerian, Sets of constant width, Pacific J. Math., 19 (1966), 13-21. doi: 10.2140/pjm.1966.19.13. [9] G. D. Chakerian and H. Groemer, Convex bodies of constant width, in Convexity and its Applications (Eds. P. M. Gruber and J. M. Wills), Birkhaüser, Basel, (1983), 49-96. [10] P. R. Chernoff, An area-width inequality for convex curves, Amer. Math. Monthly, 76 (1969), 34-35. doi: 10.2307/2316783. [11] H. Eggleston, A proof of Blaschke's theorem on the Reuleaux triangle, Quart. J. Math. Oxford, 3 (1952), 296-297. doi: 10.1093/qmath/3.1.296. [12] W. J. Firey, Lower bounds for volumes of convex bodies, J. Arch. Math., 16 (1965), 69-74. doi: 10.1007/BF01220001. [13] M. Fujiwara, Analytic proof of Blaschke's theorem on the curve of constant breadth with minimum area I and II, Proc. Imp. Acad. Japan, 3 (1927), 307-309 and 7 (1931), 300-302. doi: 10.3792/pia/1195581847. [14] M. Ghandehari, An optimal control formulation of the Blaschke-Lebesgue theorem, J. Math. Anal. Appl., 200 (1996), 322-331. doi: 10.1006/jmaa.1996.0208. [15] P. M. Gruber, Convex and Discrete Geometry, Springer-Verlag, Berlin Heidelberg, 2007. [16] E. Harrell, A direct proof of a theorem of Blaschke and Lebesgue, J. Geom. Anal., 12 (2002), 81-88. doi: 10.1007/BF02930861. [17] R. Howard, Convex bodies of constant width and constant brightness, Advances in Mathematics, 204 (2006), 241-261. doi: 10.1016/j.aim.2005.05.015. [18] H. Lebesgue, Sur le problème des isopérmètres et sur les domains de largeur constante, Bull. Soc. Math. France, C.R., 7 (1914), 72-76. [19] H. Lebesgue, Sur quelques questions des minimums, relatives aux courbes orbiformes, et sur les rapports avec le calcul de variations, J. Math. Pure Appl., 4 (1921), 67-96. [20] F. Malagoli, An Optimal Control Theory Approach to the Blaschke-Lebesgue Theorem, J. Convex Analysis, 16 (2009), 391-407. [21] K. Ou and S. L. Pan, Some remarks about closed convex curves, Pacific J. Math., 248 (2010), 393-401. doi: 10.2140/pjm.2010.248.393. [22] R. Schneider, Convex Bodies: The Brunn-Minkowski Theory, Cambridge University Press, Cambridge, 1993. doi: 10.1017/CBO9780511526282. [23] M. Sholander, On certain minimum problems in the theory of convex curves, Trans. Amer. Math. Soc., 73 (1952), 139-173. doi: 10.1090/S0002-9947-1952-0053536-4. [24] A. C. Thompson, Minkowski Geometry, Cambridge University Press, Cambridge, 1996. doi: 10.1017/CBO9781107325845.

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##### References:
 [1] H. Anciaux and N. Georgiou, The Blaschke-Lebesgue problem for constant width bodies of revolution, 2009. preprint, arXiv:0903.4284. [2] H. Anciaux and B. Guilfoyle, On the three-dimensional Blaschke-Lebesgue problem, Proc. Amer. Math. Soc., 139 (2011), 1831-1839. doi: 10.1090/S0002-9939-2010-10588-9. [3] T. Bayen, Analytical parameterization of rotors and proof of a Goldberg conjecture by optimal control theory, SIAM J. Control Optim., 47 (2008), 3007-3036. doi: 10.1137/070705325. [4] T. Bayen, T. Lachand-Robert and E. Oudet, Analytic parametrizations and volume minimization of three dimensional bodies of constant width, Arch. Ration. Mech. Anal., 186 (2007), 225-249. doi: 10.1007/s00205-007-0060-x. [5] W. Blaschke, Konvexe Bereiche gegebener konstanter Breite und kleinsten, Inhalts, Math. Ann., 76 (1915), 504-513. doi: 10.1007/BF01458221. [6] W. Blaschke, Kreis und Kugel, $2^{nd}$ edition, Gruyter, Berlin, 1956. [7] S. Campi, A. Colesanti and P. Gronchi, Minimum problems for volumes of convex bodies, in Partial Diferential Equations and Applications (eds. P. Marcellini, G. Talenti, and E. Visintin), Marcel-Dekker, New York, 177 (1996), 43-55. [8] G. D. Chakerian, Sets of constant width, Pacific J. Math., 19 (1966), 13-21. doi: 10.2140/pjm.1966.19.13. [9] G. D. Chakerian and H. Groemer, Convex bodies of constant width, in Convexity and its Applications (Eds. P. M. Gruber and J. M. Wills), Birkhaüser, Basel, (1983), 49-96. [10] P. R. Chernoff, An area-width inequality for convex curves, Amer. Math. Monthly, 76 (1969), 34-35. doi: 10.2307/2316783. [11] H. Eggleston, A proof of Blaschke's theorem on the Reuleaux triangle, Quart. J. Math. Oxford, 3 (1952), 296-297. doi: 10.1093/qmath/3.1.296. [12] W. J. Firey, Lower bounds for volumes of convex bodies, J. Arch. Math., 16 (1965), 69-74. doi: 10.1007/BF01220001. [13] M. Fujiwara, Analytic proof of Blaschke's theorem on the curve of constant breadth with minimum area I and II, Proc. Imp. Acad. Japan, 3 (1927), 307-309 and 7 (1931), 300-302. doi: 10.3792/pia/1195581847. [14] M. Ghandehari, An optimal control formulation of the Blaschke-Lebesgue theorem, J. Math. Anal. Appl., 200 (1996), 322-331. doi: 10.1006/jmaa.1996.0208. [15] P. M. Gruber, Convex and Discrete Geometry, Springer-Verlag, Berlin Heidelberg, 2007. [16] E. Harrell, A direct proof of a theorem of Blaschke and Lebesgue, J. Geom. Anal., 12 (2002), 81-88. doi: 10.1007/BF02930861. [17] R. Howard, Convex bodies of constant width and constant brightness, Advances in Mathematics, 204 (2006), 241-261. doi: 10.1016/j.aim.2005.05.015. [18] H. Lebesgue, Sur le problème des isopérmètres et sur les domains de largeur constante, Bull. Soc. Math. France, C.R., 7 (1914), 72-76. [19] H. Lebesgue, Sur quelques questions des minimums, relatives aux courbes orbiformes, et sur les rapports avec le calcul de variations, J. Math. Pure Appl., 4 (1921), 67-96. [20] F. Malagoli, An Optimal Control Theory Approach to the Blaschke-Lebesgue Theorem, J. Convex Analysis, 16 (2009), 391-407. [21] K. Ou and S. L. Pan, Some remarks about closed convex curves, Pacific J. Math., 248 (2010), 393-401. doi: 10.2140/pjm.2010.248.393. [22] R. Schneider, Convex Bodies: The Brunn-Minkowski Theory, Cambridge University Press, Cambridge, 1993. doi: 10.1017/CBO9780511526282. [23] M. Sholander, On certain minimum problems in the theory of convex curves, Trans. Amer. Math. Soc., 73 (1952), 139-173. doi: 10.1090/S0002-9947-1952-0053536-4. [24] A. C. Thompson, Minkowski Geometry, Cambridge University Press, Cambridge, 1996. doi: 10.1017/CBO9781107325845.
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