For nonnegative weights and a candidate covariance dominated entrywise by \(U\), the quadratic form is dominated by the form at \(U\); the supremum over the box is therefore attained at the upper corner. No positive-semidefiniteness of \(U\) is required, which is what makes the result usable on a bracket that is not itself a covariance matrix.

For long-only weights the squared systematic risk lies between the quadratic forms of the entrywise lower and upper brackets built from the per-asset radii.