Abstract
We introduce and study a sequence of geometric invariants for convex bodies in finite-dimensional spaces, which is in a sense dual to the sequence of mean Minkowski measures of symmetry proposed by the second author. It turns out that the sequence introduced in this paper shares many nice properties with the sequence of mean Minkowski measures, such as the sub-arithmeticity and the upper-additivity. More meaningfully, it is shown that this new sequence of geometric invariants, in contrast to the sequence of mean Minkowski measures which provides information on the shapes of lower dimensional sections of a convex body, provides information on the shapes of orthogonal projections of a convex body. The relations of these new invariants to the well-known Minkowski measure of asymmetry and their further applications are discussed as well.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Belloni A, Freund R M. On the asymmetry function of a convex set. Math Program Ser B, 2008, 111: 57–93
Groemer H. Stability theorems for two measures of symmetry. Discrete Comput Geom, 2000, 24: 301–311
Grünbaum B. Measures of symmetry for convex sets. In: Convexity: Proceedings of the Seventh Symposium in Pure Mathematics of the American Mathematical Society, vol. 7. Providence: Amer Math Soc, 1963, 233–270
Guo Q. Stability of the Minkowski measure of asymmetry for convex bodies. Discrete Comput Geom, 2005, 34: 351–362
Guo Q. On p-measure of asymmetry for convex bodies. Adv Geom, 2012, 12: 287–301
Guo Q, Guo J F, Su X L. The measures of asymmetry for coproducts of convex bodies. Pacific J Math, 2015, 276: 401–418
Guo Q, Toth G. Dual mean Minkowski measures and the grÜnbaum conjecture for affine diameters. Http://math. camden.rutgers.edu/files/Dual-Mean.pdf, 2015
Hiriart-Urruty J-B, Lemaréchal C. Fundamentals of Convex Analysis. Berlin-Heidelberg-New York: Springer-Verlag, 2001
Jin H L, Guo Q. Asymmetry of convex bodies of constant width. Discrete Comput Geom, 2012, 47: 415–423
Jin H L, Guo Q. A note on the extremal bodies of constant width for the Minkowski measure. Geom Dedicata, 2013, 164: 227–229
Jin H L, Leng G S, Guo Q. Mixed volumes and measures of asymmetry. Acta Math Sin Engl Ser, 2014, 30: 1905–1916
Klee V L. The critical set of a convex set. Amer J Math, 1953, 75: 178–188
Meyer M, Schütt C, Werner E. New affine measures of symmetry for convex bodies. Adv Math, 2011, 228: 2920–2942
Schneider R. Stability for some extremal properties of the simplex. J Geom, 2009, 96: 135–148
Schneider R. Convex Bodies: The Brunn-Minkowski Theory, 2nd ed. Cambridge: Cambridge University Press, 2014
Toth G. Simplicial intersections of a convex set and moduli for spherical minimal immersions. Michigan Math J, 2004, 52: 341–359
Toth G. On the shape of the moduli of spherical minimal immersions. Trans Amer Math Soc, 2006, 358: 2425–2446
Toth G. On the structure of convex sets with applications to the moduli of spherical minimal immersions. Beiträge Algebra Geom, 2008, 49: 491–515
Toth G. On the structure of convex sets with symmetries. Geom Dedicata, 2009, 143: 69–80
Toth G. Fine structure of convex sets from asymmetric viewpoint. Beiträge Algebra Geom, 2011, 52: 171–189
Toth G. A measure of symmetry for the moduli of spherical minimal immersions. Geom Dedicata, 2012, 160: 1–14
Toth G. Minimal simplices inscribed in a convex body. Geom Dedicata, 2014, 170: 303–318
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Guo, Q., Toth, G. Dual mean Minkowski measures of symmetry for convex bodies. Sci. China Math. 59, 1383–1394 (2016). https://doi.org/10.1007/s11425-016-5121-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11425-016-5121-x