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THE ROSENTHAL–SZASZ INEQUALITY FOR NORMED PLANES

Part of: Classical differential geometry General convexity

Published online by Cambridge University Press:  29 August 2018

VITOR BALESTRO Open the ORCID record for VITOR BALESTRO [Opens in a new window]  and
HORST MARTINI
VITOR BALESTRO*
Affiliation:
CEFET/RJ, Campus Nova Friburgo, 28635000 Nova Friburgo, Brazil email vitorbalestro@gmail.com
HORST MARTINI
Affiliation:
Fakultät für Mathematik, Technische Universität Chemnitz, 09107 Chemnitz, Germany email martini@mathematik.tu-chemnitz.de
*
✉vitorbalestro@gmail.com
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Abstract

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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.

Keywords

anti-normRosenthal–Szasz inequalitynormed planeconstant widthRadon planesupport function

MSC classification

Primary: 52A10: Convex sets in $2$ dimensions (including convex curves)
Secondary: 52A21: Finite-dimensional Banach spaces (including special norms, zonoids, etc.) 52A40: Inequalities and extremum problems 53A35: Non-Euclidean differential geometry
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 99 , Issue 1 , February 2019 , pp. 130 - 136
Copyright
© 2018 Australian Mathematical Publishing Association Inc. 

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