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  • THE ROSENTHAL–SZASZ INEQUALITY FOR NORMED PLANES

  • Part of
  • VITOR BALESTRO, HORST MARTINI
  • Journal:
    Bulletin of the Australian Mathematical Society / Volume 99 / Issue 1 / February 2019
    Published online by Cambridge University Press:
    29 August 2018, pp. 130-136
    Print publication:
    February 2019
    • Article
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  • We study the classical Rosenthal–Szasz inequality for a plane whose geometry is determined by a norm. This inequality states that the bodies of constant width have the largest perimeter among all planar convex bodies of given diameter. In the case where the unit circle of the norm is given by a Radon curve, we obtain an inequality which is completely analogous to the Euclidean case. For arbitrary norms we obtain an upper bound for the perimeter calculated in the anti-norm, yielding an analogous characterisation of all curves of constant width. To derive these results, we use methods from the differential geometry of curves in normed planes.