×

Stability result for the extremal Grünbaum distance between convex bodies. (English) Zbl 1431.52012

Summary: In [Proc. Sympos. Pure Math. 7, 233–270 (1963; Zbl 0142.20503)], B. Grünbaum introduced the following variation of the Banach-Mazur distance for arbitrary convex bodies \(K, L \subset \mathbb{R}^n: d_G(K, L) = \inf \{ |r| : K'' \subset L' \subset rK' \}\) with the infimum taken over all non-degenerate affine images \(K'\) and \(L'\) of \(K\) and \(L\) respectively. In [J. Differ. Geom. 68, No. 1, 99–119 (2004; Zbl 1120.52004)], Y. Gordon et al. proved that the maximal possible distance is equal to \(n\), confirming the conjecture of Grünbaum. In [Isr. J. Math. 183, 103–115 (2011; Zbl 1229.52005)], C. H. Jiménez and M. Naszódi asked if the equality \(d_G(K, L)=n\) implies that \(K\) or \(L\) is a simplex and they proved it under the additional assumption that one of the bodies is smooth or strictly convex. The aim of the paper is to give a stability result for a smooth case of the theorem of Jiménez and Naszódi. We prove that for each smooth convex body \(L\) there exists \(\varepsilon_0(L) >0\) such that if \(d_G(K, L) \geq (1-\varepsilon)n\) for some \(0 \leq \varepsilon \leq \varepsilon_0(L)\), then \(d(K, S_n) \leq 1 + 40n^3r (\varepsilon)\), where \(S_n\) is the simplex in \(\mathbb{R}^n\), \(r(\varepsilon)\) is a specific function of \(\varepsilon\) depending on the modulus of the convexity of the polar body of \(L\) and \(d\) is the usual Banach-Mazur distance. As a consequence, we obtain that for arbitrary convex bodies \(K, L \subset \mathbb{R}^n\) their Banach-Mazur distance is less than \(n^2 - 2^{-22}n^{-7} \).

MSC:

52A40 Inequalities and extremum problems involving convexity in convex geometry
52A20 Convex sets in \(n\) dimensions (including convex hypersurfaces)
52A27 Approximation by convex sets
PDFBibTeX XMLCite
Full Text: arXiv Link

References:

[1] K. Ball, E. A. Carlen, E. H. Lieb:Sharp uniform convexity and smoothness inequalities for trace norms, Invent. Math 115 (1994) 463-482. · Zbl 0803.47037
[2] K. Böröczky:The stability of the Rogers-Shephard inequality and of some related inequalities, Advances Math. 190 (2005) 47-76. · Zbl 1063.52002
[3] S. Brodiuk, N. Palko, A. Prymak:On Banach-Mazur distance between planar convex bodies, Aequationes Math. 92 (2018) 993-1000. · Zbl 1398.52004
[4] T. Figiel:On the moduli of convexity and smoothness, Studia Math. 56 (1976) 121-155. · Zbl 0344.46052
[5] B. Fleury, O. Guédon, G. Paouris:A stability result for mean width ofLp-centroid bodies, Advances Math. 214 (2007) 865-877. · Zbl 1132.52012
[6] E. D. Gluskin:The diameter of the Minkowski compactum is approximately equal ton(Russian), Funct. Anal. Appl. 15 (1981) 72-73. · Zbl 0469.46017
[7] Y. Gordon, A. E. Litvak, M. Meyer, A. Pajor:John’s decomposition in the general case and applications, J. Differential Geom. 68 (2004) 99-119. · Zbl 1120.52004
[8] H. Groemer:Stability of geometric inequalities, in:Handbook of Convex Geometry, North-Holland, Amsterdam (1993) 125-150. · Zbl 0789.52001
[9] B. Grünbaum:Measures of symmetry for convex sets, Proc. Sympos. Pure Math. 7 (1963) 233-270. · Zbl 0142.20503
[10] A. J. Guirao, P. Hajek:On the moduli of convexity, Proc. Amer. Math. Soc. 135 (2007) 3233-3240. · Zbl 1129.46004
[11] Q. Guo:Stability of the Minkowski measure of asymmetry for convex bodies, Discrete Comp. Geom. 34 (2005) 351-362. · Zbl 1079.52003
[12] O. Hanner:On the uniform convexity ofLpandℓp, Arkiv for Math. 3 (1956) 239-244. · Zbl 0071.32801
[13] C. H. Jiménez, M. Naszódi:On the extremal distance between two convex bodies, Isr. J. Math. 183 (2011) 103-115. · Zbl 1229.52005
[14] F. John:Extremum problems with inequalities as subsidiary conditions, Courant Anniversary Volume, Interscience (1948) 187-204. · Zbl 0034.10503
[15] M. Kiderlen:Stability results for convex bodies in geometric tomography, Indiana Univ. Math. J. 57 (2008) 1999-2038. · Zbl 1161.52004
[16] M. Lassak:Banach-Mazur distance of planar bodies, Aequationes Math. 74 (2007) 282-286. · Zbl 1137.52002
[17] K. Leichtweiss:Über die affine Exzentrizität konvexer Körper, Archiv der Mathematik 10 (1959) 187-199. · Zbl 0089.38303
[18] J. Lindenstrauss, L. Tzafriri:Classical Banach Spaces. II: Function Spaces, Springer, Berlin (1979). · Zbl 0403.46022
[19] A. Meir:On the uniform convexity ofLpspaces, Illinois J. Math. 28 (1984) 420- 424. · Zbl 0562.46014
[20] O. Palmon:The only convex body with extremal distance from the ball is the simplex, Israel J. Math. 80 (1992) 337-349. · Zbl 0774.52003
[21] M. Rudelson:Distances between non-symmetric convex bodies and the MM*estimate, Positivity 4 (2000) 161-178. · Zbl 0959.52008
[22] R. Schneider:Simplices, Educational Talks in the Research Semester on Geometric Methods in Analysis and Probability, Erwin Schrödinger Institute, Vienna (2005).
[23] R. Schneider:Stability for some extremal properties of the simplex, J. Geom. 96 (2009) 135-148. · Zbl 1201.52008
[24] M. Stephen, V. Yaskin:Stability results for sections of convex bodies, Trans. Amer. Math. Soc. 369 (2017) 6239-6261. · Zbl 1375.52004
[25] W. Stromquist:The maximum distance between two-dimensional Banach spaces, Math. Scand. 48 (1981) 205-225. · Zbl 0475.46011
[26] N. Tomczak-Jaegermann:The moduli of smoothness and convexity and the Rademacher averages of the trace classesSp(1≤p <∞), Studia Math. 50 (1974) 163-182. · Zbl 0282.46016
[27] N. Tomczak-Jaegermann:Banach-Mazur Distances and Finite-Dimensional Operator Ideals, · Zbl 0721.46004
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.