TITLE: Can closed curves have small curvature?
QUESTION [6 upvotes]: Let $\gamma$ be a smooth curve in Euclidean space of length $2\pi$ whose curvature function satisfies $-1 < k(t) < 1$.  Can $\gamma$ be closed?
This seems like it should be an easy exercise, at least in the plane, but I'm stuck...

REPLY [3 votes]: do Carmo's Differential Geometry of Curves and Surfaces mentions Fenchel's theorem on page 399:

The total curvature of a simple closed curve is at least $2\pi$

(plus something about when equality holds; this statements differs from the wikipedia page for Fenchel's theorem.)
In your case, you have that the total curvature $\int |k(s)| ds$ is $< 2\pi$. So the curve cannot be closed (if it is simple).
There is a generalization to piecewise regular closed curves in $\mathbb R^n$, known as the Fenchel-Borsuk theorem, though I can't find a handy reference right now.