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Fenchel's theorem
In differential geometry, Fenchel's theorem (Werner Fenchel, 1929) states that the average curvature of any closed convex plane curve is
\( \frac{2 \pi}{P}, \)
where P is the perimeter. More generally, for an arbitrary closed curve in space the average curvature is ( \ge \frac{2 \pi}{P} \) with equality holding only for convex plane curves.
References
W. Fenchel, Über Krümmung und Windung geschlossener Raumkurven, Math. Ann. 101 (1929), 238-252. [1]
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
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