  Let X Nv(0, {\Lambda}) be a normal vector in v dimensions, where {\Lambda} is
diagonal. With reference to the truncated distribution of X on the interior of
a v-dimensional Euclidean ball, we completely prove a variance inequality and a
covariance inequality that were recently discussed by F. Palombi and S. Toti
[J. Multivariate Anal. 122 (2013) 355-376]. These inequalities ensure the
convergence of an algorithm for the reconstruction of {\Lambda} only on the
basis of the covariance matrix of X truncated to the Euclidean ball. The
concept of monotone likelihood ratio is useful in our proofs. Moreover, we also
prove and utilize the fact that the cumulative distribution function of any
positive linear combination of independent chi-square variates is log-concave,
even though the same may not be true for the corresponding density function.
