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:

19300972

Publisher:

Mathematical Association of America

Date Published:

20170101

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 (FP7/20072013) 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 (FP7/20072013) under REA grant agreement n◦[628803].

Open access: 
yes (repository) 