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Integrate unit normal vector over unit sphere intersected with a simplicial cone

Given a set of linearly independent vectors $S = \{s_0, \ldots, s_k\} \subseteq \mathbb{R}^{n+1}$, define $$ \Delta(S) = \{c_1 s_1 + \cdots + c_k s_k \mid c_i \geq 0\} \cap \mathbb{S}^{n}, \; \Delta^\...
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