The geometry of circumscribing polygons of minimal perimeter
type: Item
instance of: scholarly article
title: The geometry of circumscribing polygons of minimal perimeter
zbMATH Open document ID: 0817.52004
DOI: 10.1007/BF01228052
published in: Journal of Geometry
publication date: 1994-05-30T00:00:00Z
review text: Let \(C\) be a closed, strictly convex curve in \(\mathbb{R}^ 2\) and \(C(n)\) the class of all \(n\)-sided polygons which circumscribe \(C\). The author establishes necessary geometric conditions for a polygon \(\prod \in C(n)\) to have minimal perimeter. As an example it is derived that for \(C\) being a circle the \(\prod^* \in C(n)\) of minimal perimeter is the regular \(n\)-gon. Next, the special cases of circumscribing triangles and quadrilaterals are investigated in great detail. Also polygons \(\prod \in C(n)\) of minimal area are considered and the following result of M. M. Day (1947) is obtained: If \(\prod \in C(n)\) has minimal area, then \(\prod = V_ 1V_ 2 \dots V_ n\), \(V_ i = i\)-th vertex of \(\prod\), touches \(C\) at the midpoints of the edges \(\overline {V_ i V_{i+1}}\), \(i = 1,2, \dots, n\).
Mathematics Subject Classification ID: 52A10
zbMATH DE Number: 558733
zbMATH Keywords: circumscribed polygons
zbMATH Keywords: minimal perimeter
zbMATH Keywords: minimal area
author: Duane DeTemple
MaRDI profile type: Publication
cites work: On polygons circumscribing a closed convex curve
cites work: Polygons Circumscribed About Closed Convex Curves
cites work: On minimum circumscribed polygons
cites work: On Triangles Circumscribing Plane Convex Sets
description: scientific article; zbMATH DE number 558733