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On Banach–Mazur distance between planar convex bodies

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Abstract

Upper estimates of the diameter and the radius of the family of planar convex bodies with respect to the Banach–Mazur distance are obtained. Namely, it is shown that the diameter does not exceed \(\tfrac{19-\sqrt{73}}{4}\approx 2.614\), which improves the previously known bound of 3, and that the radius does not exceed \(\frac{117}{70}\approx 1.671\).

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References

  1. Besicovitch, A.S.: Measure of asymmetry of convex curves. J. Lond. Math. Soc. 23, 237–240 (1948)

    Article  MathSciNet  Google Scholar 

  2. Fleischer, R., Mehlhorn, K., Rote, G., Welzl, E., Yap, C.: Simultaneous inner and outer approximation of shapes. Algorithmica 8(5–6), 365–389 (1992). 1990 Computational Geometry Symposium (Berkeley, CA, 1990)

    Article  MathSciNet  Google Scholar 

  3. Gluskin, E.D.: The diameter of the Minkowski compactum is roughly equal to \(n\). Funktsional. Anal. i Prilozhen. 15(1), 72–73 (1981). (Russian)

    Article  MathSciNet  Google Scholar 

  4. Gordon, Y., Litvak, A.E., Meyer, M., Pajor, A.: John’s decomposition in the general case and applications. J. Differ. Geom. 68(1), 99–119 (2004)

    Article  MathSciNet  Google Scholar 

  5. John, F.: Extremum Problems with Inequalities as Subsidiary Conditions, Studies and Essays Presentedto R. Courant on his 60th Birthday, January 8, 1948, pp. 187–204. Interscience Publishers, Inc., New York, NY (1948)

    Google Scholar 

  6. Lassak, M.: Banach–Mazur distance of planar convex bodies. Aequ. Math. 74(3), 282–286 (2007)

    Article  MathSciNet  Google Scholar 

  7. Lassak, M.: Approximation of convex bodies by triangles. Proc. Am. Math. Soc. 115(1), 207–210 (1992)

    Article  MathSciNet  Google Scholar 

  8. Rudelson, M.: Distances between non-symmetric convex bodies and the \(MM^\ast \)-estimate. Positivity 4(2), 161–178 (2000)

    Article  MathSciNet  Google Scholar 

  9. Stromquist, W.: The maximum distance between two-dimensional Banach spaces. Math. Scand. 48(2), 205–225 (1981)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Faculty of Computer Science and Cybernetics, Taras Shevchenko National University of Kyiv, Hlushkova Avenue 4d, Kiev, 03680, Ukraine

    Serhii Brodiuk & Nazarii Palko

  2. Department of Mathematics, University of Manitoba, Winnipeg, MB,  R3T2N2, Canada

    Andriy Prymak

Authors
  1. Serhii Brodiuk
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  2. Nazarii Palko
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  3. Andriy Prymak
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Corresponding author

Correspondence to Andriy Prymak.

Additional information

The third author was supported by NSERC of Canada Discovery Grant RGPIN 04863-15.

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Brodiuk, S., Palko, N. & Prymak, A. On Banach–Mazur distance between planar convex bodies. Aequat. Math. 92, 993–1000 (2018). https://doi.org/10.1007/s00010-018-0565-4

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  • DOI: https://doi.org/10.1007/s00010-018-0565-4

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