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Notes on the Minkowski measure, the Minkowski symmetral, and the Banach-Mazur distance
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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 $\cb/{\rm Aff}$ of the space of convex bodies modulo the affine transformations.
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 $\cb/{\rm Aff}$ of the space of convex bodies modulo the affine transformations.
참고문헌 (10)
- B. Grunbaum, "Measures of symmetry for convex sets, convexity", Proceedings of the Symposium in Pure Mathematics, VII, pp.233~270 (1963), American Mathematical Society
- Q. Guo, "Stability of the Minkowski measure of asymmetry for convex bodies", Discrete and Computational Geometry, 34, pp.351~362 (2005)[DOI]
- Q. Guo, "On asymmetry of some convex bodies", Discrete Comput. Geom., 27(2), pp.239~247 (2002)[DOI]
- Q. Guo, "Approximation of convex bodies by convex bodies", Northeast Math., 19(4), pp.323~332 (2003)
- F. John, "Extremum problems with inequalities as subsidiary conditions, Courant Anniversary Volume", pp.187~204 (1948), Interscience
- M. Lassak, "Approximation of convex bodies by centrally symmetric convex bodies", Geom. Dedicata, 72(1), pp.63~68 (1998)[DOI]
- H. Minkowski, "Gesammelte Abhandlungen" (1911)
- R. Schneider, "Stability for some extremal properties of the simplex", J. Geom., 96(1-2), pp.135~148 (2009)[DOI]
- G. Toth, "Notes on Schneider's stability estimates for convex sets", J. Geom., 104(3), pp.585~598 (2013)[DOI]
- G. Toth, "Measures of Symmetry for Convex Sets and Stability" (2015), Springer
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