I am trying to figure out what the next calculation of the "area" (or "volume" in higher dimensional analogues) using Stokes' theorem really means. Here is my thought process:
$2$-dimensional case: given a closed simple piece-wise smooth curve $C$ in $\mathbb{R}^2$ you can find out the area enclosed by $C$ using Green's theorem by choosing an orientation on $C$ and calculating
$$\text{Area} = \left\vert \dfrac{1}{2} \oint_C (x dy – y dx) \right\vert .$$
$2$-dimensional case in $\mathbb{R}^n$: When you have a piece-wise smooth non self intersecting map $\phi : S^1 \rightarrow \mathbb{R}^n$ such that $\phi (S^1)$ is contained in a $2$-dimensional plane of $\mathbb{R}^n$, you can again find the area enclosed by $\phi (S^1)$ by choosing an orthonormal coordinate system $(x_1,…,x_n)$, choosing an orientation on $S^1$ and calculating using Stokes' theorem
$$\text{Area}^2 = \sum_{1 \leq i < j \leq n} \left( \dfrac{1}{2} \oint_{\phi (S^1)} (x_i dx_j – x_j dx_i) \right)^2 .$$
Now my question is what is the meaning of calculating the above quantity when $\phi (S^1)$ is not contained in a $2$-dimensional plane. I want to use the same computation as above, but since I want the number I get to be independent on the coordinate system, I'll have to average with respect to all coordinate systems. To be explicit: let $\phi : S^1 \rightarrow \mathbb{R}^n$ be a piece-wise smooth non self intersecting map and fix an orientation on $S^1$. Let $SO_n (\mathbb{R})$ be the special orthogonal group and let $\mu$ be the normalized Haar measure on $SO_n (\mathbb{R})$ (i.e., $\mu (SO_n (\mathbb{R}) ) =1$). Define the "area" (or the "Stokes area" in lack of a better name) bounded by $\phi (S_1)$ as
$$\text{"Stokes area"}^2 = \int_{SO_n (\mathbb{R})} \left( \sum_{1 \leq i < j \leq n} \left( \dfrac{1}{2} \oint_{\gamma.\phi (S^1)} (x_i dx_j – x_j dx_i) \right)^2 \right) d \mu (\gamma) .$$
It is not hard to give analogues in any dimension to get "Stokes volume" (note that the analogue in dimension $1$, i.e., for $S^0$, comes out just the usual Euclidean distance between points).
My questions are:
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Is this quantity well-known / studied ? If so, I would very much appreciate a reference.
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What does this "Stokes area" represent geometrically (I have no intuition about it)?
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Is there a connection (say by some inequality) to the area of a minimal surface enclosed in $\phi (S^1)$?
Edit: by the answer of Will Sawin below, it is clear that we can consider only
$$\text{"Stokes area"}^2 = \sum_{1 \leq i < j \leq n} \left( \dfrac{1}{2} \oint_{\gamma.\phi (S^1)} (x_i dx_j – x_j dx_i) \right)^2 .$$
and that is quantity is always less or equal than the area of a surface enclosed by the curve. The question remains is there is a connection to the infimum of the areas of all the surfaces enclosed by the curve. In the case the curve is contained in a plane, the formula computes this minimum. But what about the general case: is there a constant (maybe depending on the dimension of $\mathbb{R}^n$) $c(n)$ such that
$$ \text{"Stokes area"} \geq c(n) (\text{infimum of the areas of surfaces enclosed by the curve}) ?$$
Best Answer
Yes, this is a lower bound on the area of a surface bounded by the curve. Parameterize the surface and apply Stokes' theorem. Your squared area is:
$$\sum_{1 \leq i < j \leq n} \left( \int_S dx_i dx_j \right)^2 = \int_S \int_S \sum_{1\leq i < j \leq n} dx_i dx_j dy_i dy_j$$
whereas the actual squared area is:
$$ \left( \int_S \sqrt{ \sum_{1 \leq i < j \leq n} \left( dx_i dx_j \right)^2} \right)^2= \int_S \int_S \sqrt{\sum_{1 \leq i < j \leq n} \left( dx_i dx_j \right)^2} \sqrt{\sum_{1 \leq i < j \leq n} \left( dy_i dy_j \right)^2}$$
which is at least as large by Cauchy-Schwartz. Equality occurs exactly when the area element of the surface is always pointed in the same direction in $\wedge^2 \mathbb R^n$ - that is, when the surface is flat.
This also shows that your definition is coordinate-invariant even before you average over $SO_n$.