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
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.
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