12 problems

Let $n\geq2$, let $\rho\in[-1,1]$, and let $3\leq k\leq n+1$. Let $\{A_i\}_{i=1}^k$ be a measurable partition of $\mathbb{R}^n$, meaning that $\bigcup_{i=1}^k A_i=\mathbb{R}^n$ and…

Let $m\geq3$, let $0<\rho<1$, and let $(\Omega_i)_{i=1}^m$ and $(\Omega_i')_{i=1}^m$ minimize the bilinear Gaussian noise-stability problem with equal prescribed measures, under th…

Let $m>3$, let $a_1,\dots,a_m>0$ satisfy $\sum_{i=1}^m a_i=1$, and let $w\in\mathbb{R}^{n+1}$ be the vector defined in the paper. Set $\overline{w}^{(i)}=w/a_i$. Consider measurabl…

Let a $q$-cluster in $\mathbb{R}^n$ be a decomposition into $q$ cells, with $q \geq 2$ and $n \geq q-1$. The model $q$-clusters are the simplicial clusters constructed in the paper…

Let a $q$-cluster be a decomposition of $mathbb{R}^n$ into pairwise disjoint Borel cells with prescribed Gaussian measure $v in operatorname{int}Delta^{(q-1)}$, where…

For a closed hypersurface in Euclidean space, let … where $P_\gamma$ is the Gaussian perimeter. CIMW's conjecture. The sphere minimizes the Gaussian entropy among closed hypersurfa…

The round-sphere conjecture. Among all compact hypersurfaces, the round sphere minimizes the entropy. The statement is an earlier formulation of the entropy-minimization claim, and…

Barthe's conjecture. If $\Omega$ minimizes this quantity, then, after a rotation, there exist $r>0$ and $0\leq k\leq n$ such that

Quadratic Symmetric Gaussian Problem. Either $B(0,r_a)$ or $B(0,r_a')^c$ achieves

Symmetric Gaussian Problem. If $\rho>0$, then either $(B(0,r_a),B(0,r_b)^c)$ or $(B(0,r_a')^c,B(0,r_b'))$ achieves

Equip $\mathbb{R}^n$ with the Gaussian metric. A hyperplane is a codimension-one affine subspace, and it is regarded as a solution of the isoperimetric problem when it separates th…

Simplicial-conical optimizer conjecture. Then $\{A_i\}_{i=1}^k$ is a simplicial conical partition. The conjecture isolates the main technical issue needed to extend the paper's res…