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

Akopyan A, Vysotsky V. 2017. On the lengths of curves passing through boundary points of a planar convex shape. The American Mathematical Monthly. 124(7), 588–596.

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Journal Article | Published | English

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Author
Akopyan, ArseniyISTA ; Vysotsky, Vladislav
Department
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.
Publishing Year
Date Published
2017-01-01
Journal Title
The American Mathematical Monthly
Volume
124
Issue
7
Page
588 - 596
ISSN
IST-REx-ID
909

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Akopyan A, Vysotsky V. On the lengths of curves passing through boundary points of a planar convex shape. The American Mathematical Monthly. 2017;124(7):588-596. doi:10.4169/amer.math.monthly.124.7.588
Akopyan, A., & Vysotsky, V. (2017). On the lengths of curves passing through boundary points of a planar convex shape. The American Mathematical Monthly. Mathematical Association of America. https://doi.org/10.4169/amer.math.monthly.124.7.588
Akopyan, Arseniy, and Vladislav Vysotsky. “On the Lengths of Curves Passing through Boundary Points of a Planar Convex Shape.” The American Mathematical Monthly. Mathematical Association of America, 2017. https://doi.org/10.4169/amer.math.monthly.124.7.588.
A. Akopyan and V. Vysotsky, “On the lengths of curves passing through boundary points of a planar convex shape,” The American Mathematical Monthly, vol. 124, no. 7. Mathematical Association of America, pp. 588–596, 2017.
Akopyan A, Vysotsky V. 2017. On the lengths of curves passing through boundary points of a planar convex shape. The American Mathematical Monthly. 124(7), 588–596.
Akopyan, Arseniy, and Vladislav Vysotsky. “On the Lengths of Curves Passing through Boundary Points of a Planar Convex Shape.” The American Mathematical Monthly, vol. 124, no. 7, Mathematical Association of America, 2017, pp. 588–96, doi:10.4169/amer.math.monthly.124.7.588.
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