A note on the variance of the square components of a normal multivariate within a Euclidean ball

We present arguments in favour of the inequalities $var(X_n^2|X \in B_v(\rho)) \le 2\lambda_n E[X_n^2|X \in B_v(\rho)]$,where$X \sim N_v(0,\Lambda)$isanormalvectorin$v\ge 1$ dimensions, with zero mean and covariance matrix $\mathrm{\Lambda }=\text{diag}(\lambda )$, and $B_v(\rho )$ is a centered $v$-dimensional Euclidean ball of square radius $\rho$. Such relations lie at the heart of an iterative algorithm, proposed in ref. [1] to perform a reconstruction of $\mathrm{\Lambda }$ from the covariance matrix of $X$ conditioned to $B_v(\rho )$. In the regime of strong truncation, i.e. for $\rho \lesssim \lambda_n$, the above inequality is easily proved, whereas it becomes harder for $\rho \gg {\lambda }_n$. Here, we expand both sides in a function series controlled by powers of ${\lambda }_n\mathrm{/}\rho$ and show that the coefficient functions of the series fulfill the inequality order by order if $\rho$ is sufficiently large. The intermediate region remains at present an open challenge.