Let $||x||$ be any norm over $\mathbb R^n$. Let $B_T$ the open ball with radius $T$ w.r.t. to our norm, i.e. all $x\in\mathbb R^n$ such that $||x||<T$. Let $n\in\mathbb N$.
How much is:$$\intop_{B_T} \sin^2 \frac {||x||\pi(n+\frac 1 2)} T dx$$ and $$\intop_{B_T} \sin^2 \frac {||x||\pi n} T dx$$
At least, how much is it with $L_2$ norm $||x||=||x||_2$?
EDIT: Note that we're actually interested in $L_2$ norm of a function. So maybe can one come with a Fouerier-Transform-Isometry argument to calculate it more easily.
