Vol. 74, No. 2, 1978

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An inequality involving the length, curvature, and torsions of a curve in Euclidean n-space

Joel Larry Weiner

Vol. 74 (1978), No. 2, 531–534

Let x be a closed nondegenerate Cn curve in En parametrized by arc length s. We prove an inequality for such x which lie in a ball of radius R. For nonplanar curves in E3 the inequality is

     2∫0Lκ2ds∫L0 τ2ds− (∫L0 κτ ds)2
L ≦ R ---------∫-L-2-----------
0 τ ds

where L is the length of x, and κ and τ are the curvature and torsion of x, respectively. Equality holds only if x is a great circle on a sphere of radius R. We also obtain from the general inequality necessary conditions on the length, curvature, and torsions of x in order that x be a closed curve or a closed curve with at most one corner.

Mathematical Subject Classification 2000
Primary: 53A05
Received: 11 July 1977
Published: 1 February 1978
Joel Larry Weiner