Pentagons inscribed in a closed convex curve
type: Item
instance of: scholarly article
title: Pentagons inscribed in a closed convex curve
zbMATH Open document ID: 0927.52002
author: V. V. Makeev
published in: Journal of Mathematical Sciences (New York)
publication date: 1998-11-01T00:00:00Z
review text: The following two theorems are proved in the paper: 1. Suppose that the sum of any two neighbouring angles of a convex pentagon \(A_1,\dots,A_5\) exceeds \(\pi.\) Let \(A_0\) be a fixed point on the boundary \(\partial K\) of a convex set \(K\subset\mathbb R^2.\) Then there exists an affine image of this pentagon, inscribed in \(K,\) such that \(A_0\) is the image of \(A_1.\) This theorem does not admit generalization to all pentagons inscribed in an ellipse but the following theorem holds: 2. Let \(A_1,\dots,A_5\) be the points of some ellipse, let \(K\subset\mathbb R^2\) be a convex set with \(C^4\)-smooth boundary \(\partial K\) of positive curvature, and let \(A_0\in\partial K\) be a fixed point of the boundary. Then there exists an affine image of the pentagon \(A_1,\dots,A_5\) such that this image is inscribed in \(K\) and \(A_0\) is the image of \(A_1.\)
Mathematics Subject Classification ID: 52A10
zbMATH DE Number: 1193408
zbMATH Keywords: convex pentagons
zbMATH Keywords: polygons inscribed in a convex set
zbMATH Keywords: extremum problems
zbMATH Keywords: extremal set theory
zbMATH Keywords: combinatorial geometry
reviewed by: Serguey M. Pokas
MaRDI profile type: Publication
full work available at URL: s10958-000-0014-4
OpenAlex ID: W156176903
cites work: Q3267885
cites work: Q4880468
DOI: 10.1007/S10958-000-0014-4
Recommended article: Polygons inscribed in a convex figure
Recommended article: Polygons inscribed in a closed curve and a three-dimensional convex body
Recommended article: On the diagonal and inscribed pentagons of a pentagon
Recommended article: Quadrangles inscribed in a closed curve
Recommended article: Non-self-affine convex pentagons
Recommended article: Inscribed and circumscribed polyhedra of a convex body
description: scientific article; zbMATH DE number 1193408