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What are some properties of the leading eigenvalue of a product of inversions in mutually tangent spheres?

Let $S_1, \ldots, S_n$ be a collection of $n \geq 4$ pairwise tangent hyperspheres in $\mathbb{R}^{n-2}$ with disjoint interiors, and $\iota_i$ be the inversion in $S_i$. Viewing the conformal group ...