7 problems

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 $\mu$ be a radially symmetric measure on the ambient Euclidean space with positive density $\rho(r)$ that is monotone decreasing in the radius. Let $B_0(r)$ be the ball of radi…

Let $B_0(r)$ be the ball of radius $r$ centered at the origin, and let $S$ range over simplices containing this ball. Let the Gaussian measure have density proportional to…

Let $\mathbb S^{n-1}$ be the unit sphere, and consider $N=n+1$ spherical caps of radius $R$ on it. The spherical-cap simplex conjecture. For every $R>0$, the maximal area of the un…

For $N=n+1$, let $v_1,\ldots,v_N\in\mathbb R^n$ satisfy $|v_1|=\dots=|v_N|=r$, and let $B_{v_i}(R)$ denote the ball of radius $R$ centered at $v_i$. The union-of-balls simplex conj…

Let $P(v_1,\ldots,v_N)=\int_{\mathbb R^n}\max_i e^{-|x-v_i|^2}\,dx$ for a code consisting of $N$ vectors in $\mathbb R^n$. The simplex conjecture. For $N=n+1$ and fixed…

Positive-correlation multi-Gaussian simplex conjecture. The standard simplex partition of $\mathbb{R}^n$ into $q$ parts maximizes the probability that $X_1,\ldots,X_k$ all fall int…