probability-statistics / High-dimensional probability
Litvak conjectured in 2018 that for every $p > 0$ the quantity $\mathbb{E}[\min_{i \le n} |g_i|^p]$, for $g \sim \mathcal{N}(0,\Sigma)$, is minimized over $n \times n$ correlation matrices by the Gram matrix of the regular simplex in $\mathbb{R}^{n-1}$. False: the matrix $\Sigma^{\cos}_{ij} = \cos(\pi(i-j)/n)$ already gives a strictly smaller value at $p = 2$, $n = 4$.
Temporal state
No reconciled state yet.
Append-only history
the paper proposes that the cosine matrix is the true minimizer for all p and n, and proves a stronger stochastic domination statement conditional on a new volumetric extension of Fejes Toth's zone conjecture
Research memory
Litvak conjectured in 2018 that for every $p > 0$ the quantity $\mathbb{E}[\min_{i \le n} |g_i|^p]$, for $g \sim \mathcal{N}(0,\Sigma)$, is minimized over $n \times n$ correlation matrices by the Gram matrix of the regular simplex in $\mathbb{R}^{n-1}$. False: the matrix $\Sigma^{\cos}_{ij} = \cos(\pi(i-j)/n)$ already gives a strictly smaller value at $p = 2$, $n = 4$.
the paper proposes that the cosine matrix is the true minimizer for all p and n, and proves a stronger stochastic domination statement conditional on a new volumetric extension of Fejes Toth's zone conjecture
Evidence graph
No public relationships recorded yet.