A Note on the Gaussian Minimum Conjecture

Abstract

Let $n\ge 2$ and $(X_i,1\le i\le n)$ be a centered Gaussian random vector. The Gaussian minimum conjecture says that $E\left({\min }_{1\le i\le n}|X_i|\right)\ge E\left({\min }_{1\le i\le n}|Y_i|\right)$, where $Y_1,\dots ,Y_n$ are independent centered Gaussian random variables with $E(X_i^2)=E(Y_i^2)$ for any $i=1,\dots ,n$. In this note, we will show that this conjecture holds if and only if $n=2$.

10 pages

References in corpus (1)

• The 4-D Gaussian Random Vector Maximum Conjecture and the 3-D Simplex Mean Width Conjecture