Abstract
In this paper we study the \(1\)-measure of asymmetry for convex bodies of constant width.
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Bayen, T., Lachand-Robert, T., Oudet, E.: Analytic parametrization of three-dimensional bodies of constant width. Arch. Ration. Mech. Anal. 186, 225–249 (2007)
Benson, R.V.: Euclidean Geometry and Convexity. McGraw-Hill, NY (1966)
Besicovitch, A.S.: Measures of asymmetry for convex curves, II. Curves of constant width. J. Lond. Math. Soc. 81–93 (1951)
Blaschke, W.: Einige Bemerkungen uber Kurven and Flachen von konstanter Breite. Ber. d. Verh. d. Sachs. Akad. Leipzig 67, 290–297 (1915) (Jbuch. 45, p. 731)
Chakerian, G.D.: Sets of constant width. Pac. J. Math. 19, 13–21 (1966)
Chakerian, G.D., Groemer, H.: Convex bodies of constant width. In: Gruber, P.M. (eds) (et al.) Convexity and its Applications, pp. 49–96. Birkhäuser Verlag, Basel (1983)
Chakerian, G.D., Stein, S.K.: Bisected chords of a convex body. Arch. Math. (Basel) 17, 561–565 (1966)
Eggleston, H.G.: Measures of asymmetry of convex curves of constant width and resricted radii of curvature. Quard. J. Math. Oxf. Ser. 3(2), 63–72 (1952)
Euler, L.: De Curvis Triangularibus. Acta Acad. Sci. Imp. Petropol. 3–30 (1778) (Opera Omnia: Series 1, 28, pp. 298–321)
Groemer, H., Wallen, L.J.: A measure of asymmetry for domains of constant width. Beitrage zur Algebra und Geom 42, 517–521 (2001)
Grünbaum, B.: Measure of symmetry for convex sets Convexity. In: Proceedings of Symposia in Pure Mathematics, vol. 7, pp. 233–270. American Math. Society, Providence (1963)
Guo, Q.: Stability of the Minkowski measure of asymmetry for convex bodies. Discret. Comput. Geom. 34, 351–362 (2005)
Guo, Q.: On p-measures of asymmetry for convex bodies. Adv. Geom. 12(2), 287–301 (2012)
Guo, Q., Kaijser, S.: On asymmetry of some convex bodies. Discret. Comput. Geom. 27, 239–247 (2002)
Heil, E., Martini, H.: Special convex bodies. In: Gruber, P.M., Wills, J.M. (eds) Handbook of Convex Geometry Gruber, pp. 347–385. North-Holland, Amsterdam (1993)
Jin, H.L., Guo, Q.: Asymmetry of convex bodies of constant width. Discret. Comput. Geom. 47, 415–423 (2012)
Jin, H.L., Guo, Q.: A note on the extremal bodies of constant width for the Minkowski measure. Geom. Dedic. 164, 227–229 (2013)
Kawohl, B., Weber, C.: Meissner’s mysterious bodies. Math. Intell. 33(3), 94–101 (2011)
Klee, V.L. Jr.: The critical set of a convex set. Am. J. Math. 75, 178–188 (1953)
Lachand-Robert, T., Oudet, E.: Bodies of constant width in arbitrary dimension. Math. Nachr. 280(7), 740–750 (2007)
Martini, H., Swanepoel, K.J.: The geometry of Minkowski spaces—a survey, part II. Expos. Math. 22, 93–144 (2004)
Meyer, M., Schutt, C., Werner, E.M.: New affine measures of symmetry for convex bodies. Adv. Math. 228, 2920–2942 (2011)
Schneider, R.: Convex Bodies: The Brunn-Minkowski Theory. Cambridge University Press, Cambridge (1993)
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The author would like to give sincere thanks to the referees for their valuable suggestions.
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The research was supported, in part, by national NSF of China No. 11271244 and No. 11271282.
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Jin, H. On the 1-measure of asymmetry for convex bodies of constant width. Beitr Algebra Geom 55, 201–206 (2014). https://doi.org/10.1007/s13366-013-0167-1
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DOI: https://doi.org/10.1007/s13366-013-0167-1