Computation of the Schläfli Function

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Abstract
The Schläfli function $f_{n}(x)$ allows to compute volume of a regular $(n-1)$ -dimensional spherical simplex of the dihedral angle $2\alpha =$ 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 $n\geq 4$ in the frequently used in applications interval $x\in [n-1,n+1]$ . 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.
Funding Information
  • Ministry of Science and Higher Education of the Russian Federation (agreement no. 075-02-2024-1430)

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