The $p$-variation seminorm (where $p \ge 1$) of a continuous curve $\alpha: [0,1] \to \mathbb{R}^2$ is defined as the supremum over all partitions $t_0 = 0 \le t_1 \le \cdots \le t_n = 1$ of: $\left(\sum_{i = 0}^{n-1}|\alpha(t_{i+1}) - \alpha(t_i)|^p\right)^{1/p}$
For $p > 2$ the $p$-variation of the curve making $n^2$ turns around a circle of radius $1/n$ goes to $0$ when $n \to +\infty$. On the other hand the area enclosed by each of these curves is exactly $\pi$.
I'm looking for such an example for $p = 2$. That is, I'm looking for a sequence of piecewise smooth closed curves whose $2$-variation goes to zero but such that the $\liminf$ of the sequence of areas enclosed by each curve is positive.
My motivation is that I'm studying the book "Differential equations driven by rough paths" (by T. Lyons, et al). They give the example above for $p > 2$ and state that one can construct such an example for $p = 2$. However I haven't been able to do so, so far.
This example would show that there is no way to define "enclosed area" as a continuous functional on the space of continuous curves with finite $2$-variation. Hence is important in motivating the theory of rough paths because it shows that attempting to extend the Young integral (which works for $1 \le p < 2$) any further is hopeless.
