On the lengths of curves passing through boundary points of a planar convex shape Journal Article


Author(s): Akopyan, Arseniy V; Vysotsky, Vladislav
Article Title: On the lengths of curves passing through boundary points of a planar convex shape
Alternate Title: The American Mathematical Monthly
Affiliation IST Austria
Abstract: We study the lengths of curves passing through a fixed number of points on the boundary of a convex shape in the plane. We show that, for any convex shape K, there exist four points on the boundary of K such that the length of any curve passing through these points is at least half of the perimeter of K. It is also shown that the same statement does not remain valid with the additional constraint that the points are extreme points of K. Moreover, the factor ½ cannot be achieved with any fixed number of extreme points. We conclude the paper with a few other inequalities related to the perimeter of a convex shape.
Keywords: Extreme points; perimeter; Convex shape; diameter; geometric inequality; upper bound for perimeter
Journal Title: The American Mathematical Monthly
Volume: 124
Issue 7
ISSN: 1930-0972
Publisher: Mathematical Association of America  
Date Published: 2017-01-01
Start Page: 588
End Page: 596
URL:
DOI: 10.4169/amer.math.monthly.124.7.588
Notes: The work of A.A. is supported by People Programme (Marie Curie Actions) of the European Union’s Seventh Framework Programme (FP7/2007-2013) under REA grant agreement n◦[291734]. The work of V.V. is supported by People Programme (Marie Curie Actions) of the European Union’s Seventh Framework Programme (FP7/2007-2013) under REA grant agreement n◦[628803].
Open access: yes (repository)
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