Computation of the Schläfli Function

The Schläfli function f n ( x ) allows to compute volume of a regular ( n − 1)-dimensional spherical simplex of the dihedral angle 2α = arcsec( x ) and it has many applications. For example, it defines conjectured upper bounds on the sphere packing problem and the kissing number problem, and a lower bound on the mean-squared error in the quantizing problem. The function is defined recursively via a first-order non-linear differential relation, that makes it difficult to compute, especially for large values of n . Here we present a method for an accurate numerical computation of the Schläfli function f n ( x ) for n ≥ 4 in the frequently used in applications interval x ∈ [ n − 1, n + 1]. The computation is based on the Chebyshev approximation of the function q n ( x ), which is related to the Schläfli function via a simple factor of an algebraic expression and regular in the interval. We also present the computation algorithm based on the method.