Bound of the surface area of subset of the unit sphere in $\mathbb{R}^3$ with no pair of orthogonal points

Question

I am trying to bound the measure of a measurable subset of the unit sphere in $\mathbb{R}^3$ that contains no pair of orthogonal points by $\frac{4\pi}{3}$.

Any help to handle this problem from a probabilistic viewpoint would be helpful.

Answer 1

Let $S$ be your set.

Let $x,y,z$ be a uniform random orthonormal basis, such that $x$, $y$ and $z$ are all uniform on the sphere (one needs to rigorously show such a process exists, it amounts to taking a uniform element of $SO(3)$).

Only one of the three can be in $S$ at once, so that $\{x\in S\}, \{y\in S\}$ and $\{z\in S\}$ are disjoint events. So $3\mu(S)/\mu(\mathbb S^2) = P(x\in S)+ P(y\in S)+P(z\in S)\leq 1$.