  In 1959 Fejes T\'oth posed a conjecture that the sum of pairwise non-obtuse
angles between $N$ unit vectors in $\mathbb S^d$ is maximized by periodically
repeated elements of the standard orthonormal basis. We obtain new improved
upper bounds for this sum, as well as for the corresponding energy integral. We
also provide several new approaches to the only settled case of the conjecture:
$d=1$.
