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Chakerian proved in this paper that a closed curve of length L in the unit ball in $\mathbb{R}^n$ has total curvature at least L.

In this later paper Chakerian gave a simpler proof and noted that equality holds iff the curve is of length $2\pi n$ and winds round the unit circle $n$ times.

Assume that the closed curve is smooth (no corners) and we are working in $\mathbb{R}^2$. My question is:

For closed curves with length $4\pi> L>2\pi$ within the unit circle in $\mathbb{R}^2$ (and hence non-convex) what is the minimal total curvature and for which curves is this attained?

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