It was known in antiquity that the sum of the three angles of a triangle equals π. Surprisingly, it was not until 1952 that the corresponding question for a tetrahedron was addressed. In that year, J.W. Gaddum proved that the sum of the four solid angles in a tetrahedron lies within the interval of 0,2π and those lower and upper bounds are the best possible. In 2020, H. Katsuura showed that 2π was unachievable. In this paper, we generalize these results to show that for a non-degenerate n-simplex in Rn with n≥3, the solid angles at the vertices add up to a positive number that is less than one-half the (n−1)-dimensional area of the unit sphere in Rn. We also show that there are examples for which the sum can be made arbitrarily close to the extreme values of 0 and one-half the (n−1)-dimensional area of the unit sphere in Rn.
Parks et al. (Fri,) studied this question.