2 problems

Let $\mathbb{G}^n$ denote Gaussian space, and let $\Omega^{k+1}[\mathbb{G}^n]$ be a standard simplicial $k$-bubble, meaning a $(k+1)$-partition obtained by translating a centered s…

Let $q$ satisfy $2\leq q\leq n+2$, and let $\Omega$ be a $q$-cluster on $\mathbb{S}^n$ with prescribed volume $V(\Omega)=v\in\operatorname{int}\Delta^{(q-1)}_{1}$. A standard bubbl…