Let $U, V \in \mathbb{R}^n$ are uniformly random orthogonal unit vectors, i.e. $||U||_2=||V||_2=1$ w.p. $1$, $U^{T}V=0$ w.p. $1$ and $(OU,OV)$ is distributed same as $(U,V)$ for all orthogonal $n \times n$ matrices $O$.
Consider the subset of the plane $$ \mathcal{P}_n := \left\{ (z^{T}U, z^{T}V) : z \in [-1,1]^n \right\}.$$ I am asked to show that for large $n$, $\mathcal{P}_n$ is close to a disc in the sense that $R_n/r_n$ converges in probability to $1$, where $$ R_n := \sup \left\{||z||_2 : z \in \mathcal{P}_n \right\}, \; r_n := \inf \left\{||z||_2 : z \in \mathbb{R}^2 \setminus \mathcal{P}_n \right\}.$$
It is clear that $R_n \geq r_n$. But I am not sure how to approach the other side. I hope convexity of the set $\mathcal{P}_n$ will be crucial and may be the representation of $(U,V)$ by Gaussian vectors might be helpful. Any suggestion is welcome.