Integral inequality with sines I am trying to show that there exists some constant $\mathcal{C}>0$ such that: $$\mathcal{C}\leq \int_0^1 |\sin (2\pi n x)-\sin (2\pi m x)|\;dx$$
For all distinct $m,n\in\mathbb{N}$. The constant appears to be a little above $4/5$. I have the feeling I am lacking knowledge of a certain integral inequality though. Maybe knowledge of the constant would shed light on how to proceed. Any ideas would be appreciated.

An Idea: If you put $$A=\frac{1}{2}|\sin(2\pi nx)-\sin(2\pi mx)|$$ then you have $0\leq A\leq 1$. Hence $A\geq A^2$, and you can integrate $\displaystyle \int_0^1 A^2dx$. (I have not done the computations)
