These questions come from the reading of the following article: Milman, E., & Neeman, J. (2022). The Gaussian Double-Bubble and Multi-Bubble Conjectures. Annals of Mathematics, 195(1), 89-206. (https://arxiv.org/pdf/1805.10961.pdf)
We are working on Euclidean space $\mathbb{R}^n$, with gaussian measure $\gamma = \gamma^n$.
Let us define a 'simplicial cluster' $\Omega$, obtained as the Voronoi cells of q equidistant points in $(\mathbb{R}^n,| \cdot|)$. More formally, $\Omega$ is a partition with $q$ components, where $\Omega_i = \text{int} \{x \in \mathbb{R}^n : \min_j | x-x_j| = | x-x_i| \ \} $.
In the article, it is noted that: when $q=4$ and $n=3$ (right example in Figure 2), $\Omega$ consists of 6 two-dimensional sectors meeting in threes at $120°$ angles along 4 half-lines, which in turn all meet at the origin in $\cos^{−1}(−1/3) ≃ 109°$ angles (p. 27).
My questions are the following:
How do they derive the $\cos^{−1}(−1/3)$ as the angle between half-lines?
In the general case $q=n+1$, what can we say about the geometry of an interface $\Sigma_{ij} = \partial (\Omega_i) \cap \partial (\Omega_j)$? In the case where $\gamma(\Omega_i) = \frac{1}{q}$ for all $i$, can we get the perimeter of each interface?
