Assume we have a zero-mean multivariate normal distribution $p(x) = N_n(0,\Sigma)$ with a diagonal covariance matrix $\Sigma$ in $n$ dimensions, where $\Sigma_{ii} > 0$ for all $i \in [1,n]$.
Assume we have a second distribution: $$\tilde{p}(x) = \begin{cases} \frac{1}{C} p(x),& \text{if } ||x||_2 \leq R\\ 0, & \text{otherwise} \end{cases}$$
To make sure this is a valid probability distribution we choose $C$ such that $\int \tilde{p}(x) = 1$. Essentially, we have restricted the normal distribution to an $n$-dimensional sphere centered at the origin with radius $R$, and have normalized this distribution so it is a valid PDF.
Let $\mathbb{E}_p$ and $\mathbb{E}_\tilde{p}$ be the expectation with respect to $p(x)$ and $\tilde{p}(x)$. I'm trying to prove that: $$\mathbb{E}_{\tilde{p}} ~x_i^2 ~\leq \mathbb{E}_p~ x_i^2$$ This seems trivial to me, however, proving this seems quite difficult.