Journal of Convex Analysis 22 (2015), No. 3, 853--858
Copyright Heldermann Verlag 2015
A Sharp Quantitative Estimate for the Perimeters of Convex Sets in the Plane
Menita Carozza
Dip. di Ingegneria, Universitį del Sannio, Corso Garibaldi, 82100 Benvento, Italy
carozza@unisannio.it
Flavia Giannetti
Dip. di Matematica e Applicazioni, Universitą di Napoli "Federico II", Via Cintia, 80126 Napoli, Italy
giannett@unina.it
Francesco Leonetti
Dip. di Ingegneria e Scienze dell' Informazione e Matematica, Universitą di L'Aquila, Via Vetoio, 67100 L'Aquila, Italy
leonetti@univaq.it
Antonella Passarelli di Napoli
Dip. di Matematica e Applicazioni, Universitą di Napoli "Federico II", Via Cintia, 80126 Napoli, Italy
antpassa@unina.it
[Abstract-pdf]
Let $E \subset B \subset \mathbb R^2$ be bounded, convex sets. The monotonicity of the perimeters holds, i.e. $\mathcal{H}^{1} (\partial E) \leqslant \mathcal{H}^{1}(\partial B)$. Here we give a quantitative estimate of the difference of the perimeters: it shows how much the perimeter of $B$ increases when $B$ becomes larger and larger with respect to $E$. We give an example showing that our estimate is sharp.
Keywords: Convex sets, perimeters, Hausdorff distance.
MSC: 52A10
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