This paper addresses the computation of normalized solid angle measure of polyhedral cones. This is well understood in dimensions two and three. For higher dimensions, assuming that a positive-definite criterion is met, the measure can be computed via a multivariable hypergeometric series. We present two decompositions of full-dimensional simplicial cones into finite families of cones satisfying the positive-definite criterion, enabling the use of the hypergeometric series to compute the solid angle measure of any polyhedral cone. Additionally, our second decomposition method yields cones with a special tridiagonal structure, reducing the number of required coordinates for the hypergeometric series formula. Furthermore, we investigate the convergence of the hypergeometric series for this case. Our findings provide a powerful tool for computing solid angle measures in high-dimensional spaces.

翻译：本文研究多面体锥的归一化立体角测度计算问题。在二维和三维空间中，该计算已有充分研究。对于更高维度，在满足正定判据的条件下，该测度可通过多元超几何级数进行计算。我们提出了两种将全维单纯锥分解为满足正定判据的有限锥族的方法，从而能够利用超几何级数计算任意多面体锥的立体角测度。此外，第二种分解方法生成的锥具有特殊的三对角结构，可减少超几何级数公式所需的坐标数量。最后，我们研究了该情形下超几何级数的收敛性。本文研究成果为高维空间中立体角测度的计算提供了有力工具。