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arXiv · 1605.07997

On the Lengths of Curves Passing through Boundary Points of a Planar Convex Shape

Arseniy Akopyan·Vladislav Vysotsky
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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 $\frac12$ cannot be achieved with any fixed number of extreme points. We conclude the paper with few other inequalities related to the perimeter of a convex shape.

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Arseniy Akopyan, Vladislav Vysotsky. 2016-05-25. On the Lengths of Curves Passing through Boundary Points of a Planar Convex Shape. https://arxiv.org/abs/1605.07997

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