# 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 & 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
 [1] Murat Adivar, Shu-Cherng Fang. Convex optimization on mixed domains. Journal of Industrial & Management Optimization, 2012, 8 (1) : 189-227. doi: 10.3934/jimo.2012.8.189 [2] Denis Gaidashev, Tomas Johnson. Spectral properties of renormalization for area-preserving maps. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3651-3675. doi: 10.3934/dcds.2016.36.3651 [3] Tomasz Dobrowolski. The dynamics of the kink in curved large area Josephson junction. Discrete & Continuous Dynamical Systems - S, 2011, 4 (5) : 1095-1105. doi: 10.3934/dcdss.2011.4.1095 [4] Simion Filip. Tropical dynamics of area-preserving maps. Journal of Modern Dynamics, 2019, 14: 179-226. doi: 10.3934/jmd.2019007 [5] Antonino Morassi, Edi Rosset, Sergio Vessella. Estimating area of inclusions in anisotropic plates from boundary data. Discrete & Continuous Dynamical Systems - S, 2013, 6 (2) : 501-515. doi: 10.3934/dcdss.2013.6.501 [6] Dirk Aeyels, Filip De Smet, Bavo Langerock. Area contraction in the presence of first integrals and almost global convergence. Discrete & Continuous Dynamical Systems - A, 2007, 18 (1) : 135-157. doi: 10.3934/dcds.2007.18.135 [7] Mário Bessa, César M. Silva. Dense area-preserving homeomorphisms have zero Lyapunov exponents. Discrete & Continuous Dynamical Systems - A, 2012, 32 (4) : 1231-1244. doi: 10.3934/dcds.2012.32.1231 [8] Miroslav KolÁŘ, Michal BeneŠ, Daniel ŠevČoviČ. Area preserving geodesic curvature driven flow of closed curves on a surface. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3671-3689. doi: 10.3934/dcdsb.2017148 [9] Hans Koch. On hyperbolicity in the renormalization of near-critical area-preserving maps. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 7029-7056. doi: 10.3934/dcds.2016106 [10] Jingzhi Yan. Existence of torsion-low maximal isotopies for area preserving surface homeomorphisms. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4571-4602. doi: 10.3934/dcds.2018200 [11] Giovanni Forni. The cohomological equation for area-preserving flows on compact surfaces. Electronic Research Announcements, 1995, 1: 114-123. [12] Sangkyu Baek, Bong Dae Choi. Performance analysis of power save mode in IEEE 802.11 infrastructure wireless local area network. Journal of Industrial & Management Optimization, 2009, 5 (3) : 481-492. doi: 10.3934/jimo.2009.5.481 [13] Luis F. Gordillo. Optimal sterile insect release for area-wide integrated pest management in a density regulated pest population. Mathematical Biosciences & Engineering, 2014, 11 (3) : 511-521. doi: 10.3934/mbe.2014.11.511 [14] Daniel N. Dore, Andrew D. Hanlon. Area preserving maps on $\boldsymbol{S^2}$: A lower bound on the $\boldsymbol{C^0}$-norm using symplectic spectral invariants. Electronic Research Announcements, 2013, 20: 97-102. doi: 10.3934/era.2013.20.97 [15] Peter Giesl, James McMichen. Determination of the area of exponential attraction in one-dimensional finite-time systems using meshless collocation. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1835-1850. doi: 10.3934/dcdsb.2018094 [16] Marie Henry, Danielle Hilhorst, Masayasu Mimura. A reaction-diffusion approximation to an area preserving mean curvature flow coupled with a bulk equation. Discrete & Continuous Dynamical Systems - S, 2011, 4 (1) : 125-154. doi: 10.3934/dcdss.2011.4.125 [17] Denis Gaidashev, Tomas Johnson. Dynamics of the universal area-preserving map associated with period-doubling: Stable sets. Journal of Modern Dynamics, 2009, 3 (4) : 555-587. doi: 10.3934/jmd.2009.3.555 [18] Guido De Philippis, Antonio De Rosa, Jonas Hirsch. The area blow up set for bounded mean curvature submanifolds with respect to elliptic surface energy functionals. Discrete & Continuous Dynamical Systems - A, 2019, 0 (0) : 1-28. doi: 10.3934/dcds.2019243 [19] Adil Bagirov, Sona Taheri, Soodabeh Asadi. A difference of convex optimization algorithm for piecewise linear regression. Journal of Industrial & Management Optimization, 2019, 15 (2) : 909-932. doi: 10.3934/jimo.2018077 [20] Zhongliang Deng, Enwen Hu. Error minimization with global optimization for difference of convex functions. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 1027-1033. doi: 10.3934/dcdss.2019070

2018 Impact Factor: 1.008

## Metrics

• HTML views (0)
• Cited by (0)

• on AIMS