On a multivariate normal probability minimization problem (Q1083800)

Given a 2k component (column) vector \(z=(x',y')'\), \(x'=(x_ 1,...,x_ k)\), \(y'=(y_ 1,...,y_ k)\), having a 2k variate multivariate normal distribution with mean vector zero, and an unknown 2k\(\times 2k\) positive definite correlation matrix \(\Sigma\), \textit{G. Shafer} and \textit{I. Olkin} [J. Am. Stat. Assoc. 78, 674-678 (1983; Zbl 0534.62076)] conjecture that \[ P(x,y)=P\{| x_ 1| <c,...,| x_ k| <c,| y_ 1| <c,...,| y_ k| <c\} \] is minimized when \(\Sigma\) has a certain pattern. This conjecture is proved to be correct. A known inequality is generalized.