Let $\zeta=e^{2\pi i/n}$, where $n\geq3$. Let $||\cdot||$ be the norm induced by the complex inner product $\langle\cdot,\cdot\rangle$.
Then $\langle x,y\rangle=\frac{1}{n}\sum_{k=1}^{n}||x+\zeta^ky||^2\zeta^k,$ as can easily be checked.
The analogous equation $\langle x,y\rangle=\frac{1}{2\pi}\int_{0}^{2\pi}||x+e^{i\theta}y||^2e^{i\theta}d\theta$
also holds.
My question is: What do these mean?
It seems to suggest that $\langle x,y\rangle$ can be viewed as a weighted average of the norms of points evenly spaced on the circumference of a circle of radius $||y||$ centered at $x$; however, the significance of the weights eludes me. Are they necessary to compensate for the effect of "rotating" $y$?