article
On the lengths of curves passing through boundary points of a planar convex shape
published
yes
Arseniy
Akopyan
author 430D2C90-F248-11E8-B48F-1D18A9856A870000-0002-2548-617X
Vladislav
Vysotsky
author
HeEd
department
International IST Postdoc Fellowship Programme
project
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 &#xbd; 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.
Mathematical Association of America2017
eng
The American Mathematical Monthly
00029890
1605.07997
00041394730000210.4169/amer.math.monthly.124.7.588
1247588 - 596
Akopyan, Arseniy, and Vladislav Vysotsky. “On the Lengths of Curves Passing through Boundary Points of a Planar Convex Shape.” <i>The American Mathematical Monthly</i>, vol. 124, no. 7, Mathematical Association of America, 2017, pp. 588–96, doi:<a href="https://doi.org/10.4169/amer.math.monthly.124.7.588">10.4169/amer.math.monthly.124.7.588</a>.
A. Akopyan and V. Vysotsky, “On the lengths of curves passing through boundary points of a planar convex shape,” <i>The American Mathematical Monthly</i>, 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. <i>The American Mathematical Monthly</i>. Mathematical Association of America. <a href="https://doi.org/10.4169/amer.math.monthly.124.7.588">https://doi.org/10.4169/amer.math.monthly.124.7.588</a>
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 A, Vysotsky V. On the lengths of curves passing through boundary points of a planar convex shape. <i>The American Mathematical Monthly</i>. 2017;124(7):588-596. doi:<a href="https://doi.org/10.4169/amer.math.monthly.124.7.588">10.4169/amer.math.monthly.124.7.588</a>
A. Akopyan, V. Vysotsky, The American Mathematical Monthly 124 (2017) 588–596.
Akopyan, Arseniy, and Vladislav Vysotsky. “On the Lengths of Curves Passing through Boundary Points of a Planar Convex Shape.” <i>The American Mathematical Monthly</i>. Mathematical Association of America, 2017. <a href="https://doi.org/10.4169/amer.math.monthly.124.7.588">https://doi.org/10.4169/amer.math.monthly.124.7.588</a>.
9092018-12-11T11:49:09Z2023-10-17T11:24:57Z