Abstract

In 1963 Grünbaum introduced the following variation of the Banach-Mazur distance for arbitrary convex bodies K,L⊂RnK, L \subset \mathbb{R}^n: dG(K,L)=inf⁡{∣r∣:K′⊂L′⊂rK′}d_G(K, L) = \inf \{ |r| : K' \subset L' \subset rK' \} with the infimum taken over all non-degenerate affine images K′K' and L′L' of KK and LL respectively. In 2004 Gordon, Litvak, Meyer and Pajor proved that the maximal possible distance is equal to nn, confirming the conjecture of Grünbaum. In 2011 Jiménez and Naszódi asked if the equality dG(K,L)=nd_G(K, L)=n implies that KK or LL 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 LL there exists ε0(L)>0\varepsilon_0(L) >0 such that if dG(K,L)≥(1−ε)nd_G(K, L) \geq (1-\varepsilon)n for some 0≤ε≤ε0(L)0 \leq \varepsilon \leq \varepsilon_0(L), then d(K,Sn)≤1+40n3r(ε)d(K, S_n) \leq 1 + 40n^3r (\varepsilon), where SnS_n is the simplex in Rn\mathbb{R}^n, r(ε)r(\varepsilon) is a specific function of ε\varepsilon depending on the modulus of the convexity of the polar body of LL and dd is the usual Banach-Mazur distance. As a consequence, we obtain that for arbitrary convex bodies K,L⊂RnK, L \subset \mathbb{R}^n their Banach-Mazur distance is less than n2−2−22n−7n^2 - 2^{-22}n^{-7}.

Author information

Contact details are reproduced from the original publication and may be historical.

Tomasz Kobos

Faculty of Mathematics and Comp. Science, Jagiellonian University, 30-348 Krakow, Poland

Tomasz.Kobos@im.uj.edu.pl

Keywords

  • Banach-Mazur distance
  • Grünbaum distance
  • convex body
  • stability
  • John's decomposition

Mathematics Subject Classification

52A40, 52A20, 52A27

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Suggested citation

T. Kobos. “Stability Result for the Extremal Grünbaum Distance Between Convex Bodies.” Journal of Convex Analysis 26 (2019), No. 4, 1277–1296. https://doi.org/10.68381/jca26067

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Published by Heldermann Verlag, 2019. Rights now held by Banach Press LLC.