Please use this identifier to cite or link to this item: http://hdl.handle.net/10077/4769
Title: On the perimeter deviation of a convex disc from a polygon
Authors: Florian, August
Issue Date: 1992
Publisher: Università degli Studi di Trieste. Dipartimento di Scienze Matematiche
Source: August Florian, “On the perimeter deviation of a convex disc from a polygon”, in: Rendiconti dell’Istituto di Matematica dell’Università di Trieste. An International Journal of Mathematics, 24 (1992), pp. 177-191.
Series/Report no.: Rendiconti dell’Istituto di Matematica dell’Università di Trieste. An International Journal of Mathematics
24 (1992)
Abstract: Nel piano siano C$_{1}$ e C$_{2}$ due insiemi compatti e convessi. Indichiamo con $\rho$$^{P}$(C$_{1}$ e C$_{2}$) la distanza tra loro nella metrica L$_{1}$. Si denota con P$_{n}$ un qualunque poligono convesso di n vertici al massimo. Fissato un convesso C, esiste un poligono P$_{n}$ = P$_{n}$(C) minimante la distanza $\rho$$^{P}$ (C, P$_{n}$). In questo lavoro studiamo alcune proprietà di tale P$_{n}$(C). Se l'insieme C ha il perimetro p, si prova che \[ \rho^{P}\left(C,P_{n}\left(C\right)\right)\leq p\left(1-\frac{2n}{\pi}\arcsin\left(\frac{1}{2}\sin\frac{\pi}{n}\right)\right). \] L'uguaglianza vale se C è un cerchio.
Let C$_{1}$ and C$_{2}$ be two compact convex subsets of the plane. We denote by $\rho$$^{P}$(C$_{1}$ e C$_{2}$) the distance between C$_{1}$ and C$_{2}$ determined by the L$_{1}$ metric. Let P$_{n}$ be any convex polygon with at most n vertices. Given a convex set C, there's a polygon P$_{n}$ = P$_{n}$(C) minimizing the distance $\rho$$^{P}$ (C, P$_{n}$). In this paper we study some properties of P$_{n}$(C). If the set C has the perimeter p, we prove that \[ \rho^{P}\left(C,P_{n}\left(C\right)\right)\leq p\left(1-\frac{2n}{\pi}\arcsin\left(\frac{1}{2}\sin\frac{\pi}{n}\right)\right). \] Equality holds if C is a circle.
URI: http://hdl.handle.net/10077/4769
ISSN: 0049-4704
Appears in Collections:Rendiconti dell'Istituto di matematica dell'Università di Trieste: an International Journal of Mathematics vol.24 (1992)

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