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 & 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, (). Google Scholar

[2]

H. Anciaux and B. Guilfoyle, On the three-dimensional Blaschke-Lebesgue problem,, Proc. Amer. Math. Soc., 139 (2011), 1831. doi: 10.1090/S0002-9939-2010-10588-9. Google Scholar

[3]

T. Bayen, Analytical parameterization of rotors and proof of a Goldberg conjecture by optimal control theory,, SIAM J. Control Optim., 47 (2008), 3007. doi: 10.1137/070705325. Google Scholar

[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. doi: 10.1007/s00205-007-0060-x. Google Scholar

[5]

W. Blaschke, Konvexe Bereiche gegebener konstanter Breite und kleinsten,, Inhalts, 76 (1915), 504. doi: 10.1007/BF01458221. Google Scholar

[6]

W. Blaschke, Kreis und Kugel,, $2^{nd}$ edition, (1956). Google Scholar

[7]

S. Campi, A. Colesanti and P. Gronchi, Minimum problems for volumes of convex bodies,, in Partial Diferential Equations and Applications (eds. P. Marcellini, 177 (1996), 43. Google Scholar

[8]

G. D. Chakerian, Sets of constant width,, Pacific J. Math., 19 (1966), 13. doi: 10.2140/pjm.1966.19.13. Google Scholar

[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), (1983), 49. Google Scholar

[10]

P. R. Chernoff, An area-width inequality for convex curves,, Amer. Math. Monthly, 76 (1969), 34. doi: 10.2307/2316783. Google Scholar

[11]

H. Eggleston, A proof of Blaschke's theorem on the Reuleaux triangle,, Quart. J. Math. Oxford, 3 (1952), 296. doi: 10.1093/qmath/3.1.296. Google Scholar

[12]

W. J. Firey, Lower bounds for volumes of convex bodies,, J. Arch. Math., 16 (1965), 69. doi: 10.1007/BF01220001. Google Scholar

[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. doi: 10.3792/pia/1195581847. Google Scholar

[14]

M. Ghandehari, An optimal control formulation of the Blaschke-Lebesgue theorem,, J. Math. Anal. Appl., 200 (1996), 322. doi: 10.1006/jmaa.1996.0208. Google Scholar

[15]

P. M. Gruber, Convex and Discrete Geometry,, Springer-Verlag, (2007). Google Scholar

[16]

E. Harrell, A direct proof of a theorem of Blaschke and Lebesgue,, J. Geom. Anal., 12 (2002), 81. doi: 10.1007/BF02930861. Google Scholar

[17]

R. Howard, Convex bodies of constant width and constant brightness,, Advances in Mathematics, 204 (2006), 241. doi: 10.1016/j.aim.2005.05.015. Google Scholar

[18]

H. Lebesgue, Sur le problème des isopérmètres et sur les domains de largeur constante,, Bull. Soc. Math. France, 7 (1914), 72. Google Scholar

[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. Google Scholar

[20]

F. Malagoli, An Optimal Control Theory Approach to the Blaschke-Lebesgue Theorem,, J. Convex Analysis, 16 (2009), 391. Google Scholar

[21]

K. Ou and S. L. Pan, Some remarks about closed convex curves,, Pacific J. Math., 248 (2010), 393. doi: 10.2140/pjm.2010.248.393. Google Scholar

[22]

R. Schneider, Convex Bodies: The Brunn-Minkowski Theory,, Cambridge University Press, (1993). doi: 10.1017/CBO9780511526282. Google Scholar

[23]

M. Sholander, On certain minimum problems in the theory of convex curves,, Trans. Amer. Math. Soc., 73 (1952), 139. doi: 10.1090/S0002-9947-1952-0053536-4. Google Scholar

[24]

A. C. Thompson, Minkowski Geometry,, Cambridge University Press, (1996). doi: 10.1017/CBO9781107325845. Google Scholar

show all references

References:
[1]

H. Anciaux and N. Georgiou, The Blaschke-Lebesgue problem for constant width bodies of revolution,, 2009. preprint, (). Google Scholar

[2]

H. Anciaux and B. Guilfoyle, On the three-dimensional Blaschke-Lebesgue problem,, Proc. Amer. Math. Soc., 139 (2011), 1831. doi: 10.1090/S0002-9939-2010-10588-9. Google Scholar

[3]

T. Bayen, Analytical parameterization of rotors and proof of a Goldberg conjecture by optimal control theory,, SIAM J. Control Optim., 47 (2008), 3007. doi: 10.1137/070705325. Google Scholar

[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. doi: 10.1007/s00205-007-0060-x. Google Scholar

[5]

W. Blaschke, Konvexe Bereiche gegebener konstanter Breite und kleinsten,, Inhalts, 76 (1915), 504. doi: 10.1007/BF01458221. Google Scholar

[6]

W. Blaschke, Kreis und Kugel,, $2^{nd}$ edition, (1956). Google Scholar

[7]

S. Campi, A. Colesanti and P. Gronchi, Minimum problems for volumes of convex bodies,, in Partial Diferential Equations and Applications (eds. P. Marcellini, 177 (1996), 43. Google Scholar

[8]

G. D. Chakerian, Sets of constant width,, Pacific J. Math., 19 (1966), 13. doi: 10.2140/pjm.1966.19.13. Google Scholar

[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), (1983), 49. Google Scholar

[10]

P. R. Chernoff, An area-width inequality for convex curves,, Amer. Math. Monthly, 76 (1969), 34. doi: 10.2307/2316783. Google Scholar

[11]

H. Eggleston, A proof of Blaschke's theorem on the Reuleaux triangle,, Quart. J. Math. Oxford, 3 (1952), 296. doi: 10.1093/qmath/3.1.296. Google Scholar

[12]

W. J. Firey, Lower bounds for volumes of convex bodies,, J. Arch. Math., 16 (1965), 69. doi: 10.1007/BF01220001. Google Scholar

[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. doi: 10.3792/pia/1195581847. Google Scholar

[14]

M. Ghandehari, An optimal control formulation of the Blaschke-Lebesgue theorem,, J. Math. Anal. Appl., 200 (1996), 322. doi: 10.1006/jmaa.1996.0208. Google Scholar

[15]

P. M. Gruber, Convex and Discrete Geometry,, Springer-Verlag, (2007). Google Scholar

[16]

E. Harrell, A direct proof of a theorem of Blaschke and Lebesgue,, J. Geom. Anal., 12 (2002), 81. doi: 10.1007/BF02930861. Google Scholar

[17]

R. Howard, Convex bodies of constant width and constant brightness,, Advances in Mathematics, 204 (2006), 241. doi: 10.1016/j.aim.2005.05.015. Google Scholar

[18]

H. Lebesgue, Sur le problème des isopérmètres et sur les domains de largeur constante,, Bull. Soc. Math. France, 7 (1914), 72. Google Scholar

[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. Google Scholar

[20]

F. Malagoli, An Optimal Control Theory Approach to the Blaschke-Lebesgue Theorem,, J. Convex Analysis, 16 (2009), 391. Google Scholar

[21]

K. Ou and S. L. Pan, Some remarks about closed convex curves,, Pacific J. Math., 248 (2010), 393. doi: 10.2140/pjm.2010.248.393. Google Scholar

[22]

R. Schneider, Convex Bodies: The Brunn-Minkowski Theory,, Cambridge University Press, (1993). doi: 10.1017/CBO9780511526282. Google Scholar

[23]

M. Sholander, On certain minimum problems in the theory of convex curves,, Trans. Amer. Math. Soc., 73 (1952), 139. doi: 10.1090/S0002-9947-1952-0053536-4. Google Scholar

[24]

A. C. Thompson, Minkowski Geometry,, Cambridge University Press, (1996). doi: 10.1017/CBO9781107325845. Google Scholar

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