Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

NOTES ON THE MINKOWSKI MEASURE, THE MINKOWSKI SYMMETRAL, AND THE BANACH-MAZUR DISTANCE

  • Huang, Xing (School of Mathematics and Information Science Guangzhou University)
  • Received : 2017.06.20
  • Accepted : 2017.08.11
  • Published : 2018.05.01
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper we derive some basic inequalities connecting the Minkowski measure of symmetry, the Minkowski symmetral and the Banach-Mazur distance. We then explore the geometric contents of these inequalities and shed light on the structure of the quotient 𝔅/Aff of the space of convex bodies modulo the affine transformations.

Keywords

Acknowledgement

Supported by : National Natural Science Foundation of China

References

  1. B. Grunbaum, Measures of symmetry for convex sets, convexity, Proceedings of the Symposium in Pure Mathematics, Vol. VII, 233-270, American Mathematical Society, 1963.
  2. Q. Guo, Stability of the Minkowski measure of asymmetry for convex bodies, Discrete and Computational Geometry 34 (2005), 351-362. https://doi.org/10.1007/s00454-005-1161-7
  3. Q. Guo and S. Kaijser, On asymmetry of some convex bodies, Discrete Comput. Geom. 27 (2002), no. 2, 239-247. https://doi.org/10.1007/s00454-001-0059-2
  4. Q. Guo and S. Kaijser, Approximation of convex bodies by convex bodies, Northeast Math. 19 (2003), no. 4, 323-332.
  5. F. John, Extremum problems with inequalities as subsidiary conditions, Courant Anniversary Volume, 187-204, Interscience, New York, 1948.
  6. M. Lassak, Approximation of convex bodies by centrally symmetric convex bodies, Geom. Dedicata 72 (1998), no. 1, 63-68. https://doi.org/10.1023/A:1005055415136
  7. H. Minkowski, Gesammelte Abhandlungen, Leipzig-Berlin, 1911.
  8. R. Schneider, Stability for some extremal properties of the simplex, J. Geom. 96 (2009), no. 1-2, 135-148. https://doi.org/10.1007/s00022-010-0028-0
  9. G. Toth, Notes on Schneider's stability estimates for convex sets, J. Geom. 104 (2013), no. 3, 585-598. https://doi.org/10.1007/s00022-013-0179-x
  10. G. Toth, Measures of Symmetry for Convex Sets and Stability, Springer, 2015.