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THE ROSENTHAL–SZASZ INEQUALITY FOR NORMED PLANES

Published online by Cambridge University Press:  29 August 2018

VITOR BALESTRO*
Affiliation:
CEFET/RJ, Campus Nova Friburgo, 28635000 Nova Friburgo, Brazil email [email protected]
HORST MARTINI
Affiliation:
Fakultät für Mathematik, Technische Universität Chemnitz, 09107 Chemnitz, Germany email [email protected]
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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.

Type
Research Article
Copyright
© 2018 Australian Mathematical Publishing Association Inc. 

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