= Solution
Let $c$ be the center of the containing disk and take arclength around the closed curve. The periodic function $\Phi(s)=|\gamma(s)-c|^2/2$ has a maximum. At a maximum $s_0$,
$$
0\ge\Phi''(s_0)=1+(\gamma(s_0)-c)\cdot\gamma''(s_0).
$$
By the <Cauchy-Schwarz inequality> and the containing radius, $1\le|\gamma(s_0)-c|\,|\gamma''(s_0)|\le r|k(s_0)|$. Therefore
$$
\boxed{|k(s_0)|\ge r^{-1}.}
$$
Closedness provides an interior maximum on the periodic parameter domain. Without it, the straight segment $\gamma(s)=(s,0)$ for $-r<s<r$ lies in the disk but has signed <curvature> identically zero, giving the required counterexample.
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