13 problems

Let $m\geq 3$ and $0<\rho<1$. Consider two measurable partitions $\Omega_1,\ldots,\Omega_m$ and $\Omega'_1,\ldots,\Omega'_m$ of $\mathbb{R}^{n+1}$ with…

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 $\gamma(E)$ and $P_\gamma(E)$ denote the Gaussian measure and Gaussian perimeter of a set $E\subset\mathbb R^n$. Among symmetric sets with a prescribed Gaussian measure, consid…

Let $\Omega\subseteq\mathbb{R}^{n+1}$ minimize Barthe's problem, namely the Gaussian perimeter minimization problem under a fixed Gaussian measure constraint. Write $S^k$ for the u…

Let $I_{\infty}(c)$ denote the infimum of the Gaussian perimeter among measurable sets of Gaussian measure $c\in(0,1)$, with the infimum taken over all finite dimensions. Finite-di…

Let $me3$, and let $a_1,dots,a_m>0$ satisfy … Let $\Omega_1,\dots,\Omega_m\subseteq\mathbb{R}^{n+1}$ be a minimizing partition for the Gaussian multi-bubble problem: the sets cover…

Let $m\geq 3$ and let $\Omega_{1},\ldots,\Omega_{m}\subseteq\mathbb{R}^{n+1}$ minimize the Gaussian multi-bubble problem: among measurable sets covering $\mathbb{R}^{n+1}$ with pre…

Let $obreak ktext{-clusters}$ be decompositions of $obreakmathbb{R}^n$ into $k$ cells with prescribed Gaussian measure $v in operatorname{int}Delta^{(k-1)}$, and let the total Ga…

Let $\Omega\subseteq\mathbb{R}^{n}$ be a periodic set and let $B\subseteq\mathbb{R}^{n}$ be a periodic half space. Periodic Gaussian surface-area conjecture. The Gaussian surface a…

Let $\Omega\subseteq\mathbb{R}^{n}$ be a periodic set, meaning that $\Omega+v=\Omega^{c}$ for every standard basis vector $v$ and $-\Omega=\Omega^{c}$. A periodic half space is a s…

Let $E\subset\mathbb{R}^n$ be an open set with Lipschitz boundary and Gaussian measure $\gamma(E)=\phi(s)$, where … Its Gaussian perimeter is … and define the Gaussian isoperimetri…

For $a\in(0,1)$, let $J(a,a;\rho)$ denote the Gaussian noise-stability quantity for two sets of Gaussian measure $a$, and let $\mathbb{E}_\rho J(1_A(X),1_A(Y);\rho)$ denote the cor…

Let $A\subset\mathbb{R}^n$ have Gaussian surface area less than $\phi(\Phi^{-1}(\operatorname{Pr}(A)))+\delta$. A quantitative Gaussian isoperimetric stability conjecture asserts t…