Given $a_1, \dots, a_n \in \mathbb R_{\geq0}$, we wish to choose $x_1, \dots, x_n \in \mathbb R_{\geq 0}$ that minimise $\sum_i a_ix_i^2$ with the constraint $\sum_i x_i = 1$.
This can be solved with Lagrange multipliers the tedious way. But it turns out that the optimal set of $x_i$s satisfies $a_1x_1 = \dots = a_nx_n$. Is there an intuitive justification (perhaps statistical or geometric) for why this holds? If so, the problem could be solved by simply observing that the minimum should satisfy $a_1x_1 = \dots = a_nx_n$, from which the solution falls out trivially.
(This problem arises when constructing a minimum variance fully-invested portfolio where all assets are uncorrelated, for example.)
Yes. You can see it from the Cauchy-Schwarz inequality, which asserts that $$ \left( \sum u_i v_i \right)^2 \leq \left( \sum u_i^2 \right) \left( \sum v_i^2 \right) $$ with equality iff there is some constant $C>0$ such that for all $i$, $u_i = Cv_i$.
Apply the inequality with $u_i = \sqrt{a_i}x_i$ and $v_i = \sqrt{x_i}$. The right hand side is the quantity you want to minimize: $\sum a_i x_i^2$. So this is attained when $\sqrt{a_i} x_i = C \sqrt{x_i}$, or $a_ix_i = C^2$.