14 problems

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…

Let $\gamma$ denote the standard Gaussian measure on $\mathbb R$. For $n,d\in\mathbb N$, let $\mathcal P_d(\mathbb R^n)$ be the space of polynomials on $\mathbb R^n$ of degree at m…

Let $d$ be a positive integer, let $(\mathbb{R}^d,g,\mu)$ satisfy $\operatorname{CD}(\rho,\infty)$ for some $\rho>0$, and define the centered Gaussian probability measure … where…

Let $d$ be a positive integer, let $(\mathbb{R}^d,g,\mu)$ satisfy the curvature-dimension condition $\operatorname{CD}(\rho,\infty)$ for some $\rho>0$, and let…

Let $\boldsymbol{A},\boldsymbol{B}\rhd 0$ be positive-definite covariance operators on $\mathcal{H}=\mathbb{R}^{d}$, and let $\pi_{\varepsilon}$ and $\pi_{0}$ denote the entropic a…

Gaussian $n^{-1}$ convergence conjecture.

Optimal growth conjecture. There exists $C(s)>0$ such that, for every $t\in[0,t_0)$ and every $s\in(0,\sigma]$,

Exceptional-support conjecture. Part (i) of the theorem on invariant measures already contains essentially all counterexamples to non-invariance: for…

Let $\gamma$ be the standard Gaussian measure on $\mathbb{R}^d$, and let $\mu$ be a log-Lipschitz perturbation of $\gamma$. Let $T=\nabla\varphi$ be the Brenier map transporting…

Dashti–Stuart conjecture. Assume that the conditions in Assumption $4$ hold. Then:

Let $K$ be a local field with residue-field cardinality $q$, let $L$ be a lattice in $K^d$, and let $V$ be the tropicalization of a full-dimensional Gaussian in $K^d$ with lattice…

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 $f(x,y)=e^{\psi(x,y)}$ be the normalized sum of two Gaussian densities with the same variance and different centers. The line connecting the two centers determines a perpendicu…

Let $\gamma_n$ be the Gaussian probability measure on $\mathbb{R}^n$, and let $K_1$ and $K_2$ be symmetric convex bodies in $\mathbb{R}^n$. The Gaussian Correlation Conjecture. The…