given a set of points, $T$, on the surface of $\mathbb{S}^n$, does a $t \in T$ lie in every possible hemisphere of $\mathbb{S}^n$?

Question

Consider the $n$-dimensional hypersphere, $\mathbb{S}^n$. Given a set of points, $T$, on the surface of $\mathbb{S}^n$, we wish to determine whether at least one $t \in T$ lies in (including on the boundary of) any possible hemisphere of $\mathbb{S}^n$.

• what are the most elegant sufficiency conditions?
• what are the most efficient sufficiency conditions?

In the special case in which $t, t' \in T$ are antipodal, then we are done.

As this is not a Hamiltonian path problem (so, not as obviously threatened by greedy algorithms?), a 'nearest antipodes' approach seems a good start:

1. start with the most isolated $t \in T$;
2. if its antipode $t'$ is in $T$, stop;
3. otherwise, determine the closest $t'' \in T$ to $t' \ldots$

The $n = 2$ case was presented in the 2017 World Finals of Code Jam, as a problem in 'omnicircumnavigation'.