The Sum of the Solid Angles of an n-Simplex

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.