Journal ArticleDOI
On the Lengths of Curves Passing Through Boundary Points of a Planar Convex Shape
TLDR
In this paper, the authors studied the length of curves passing through a fixed number of points on the boundary of a convex shape in the plane and showed that any curve passing through these points is at least half of the perimeter of the shape.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.read more
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Journal ArticleDOI
Large deviations of convex hulls of planar random walks and Brownian motions
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Properties of a curve whose convex hull covers a given convex body
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One property of a planar curve whose convex hull covers a given convex figure
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