Computing farthest pair of points in d dimensions
For the case $d = 3$, there exist $O(n\log n)$ algorithms due to Clarkson and Shor (randomized) and Ramos (deterministic). For higher dimensions there seem to exist only approximation algorithms, see for example a presentation of Vigneron. You can use the Johnson–Lindenstrauss lemma to reduce the problem (approximately) to dimension $O(\log n)$, in time $\tilde{O}(dn)$ (see for example Ailon and Chazelle).