= Minimum squared norm under two affine constraints
{title2=$x_i^*=(S_2-S_1a_i)/(nS_2-S_1^2)$}
For real $a_i$ not all equal, let $S_1=\sum_i a_i$, $S_2=\sum_i a_i^2$ and $\Delta=nS_2-S_1^2>0$. The unique minimizer of $\sum_i x_i^2$ under $\sum_i x_i=1$, $\sum_i a_ix_i=0$ is $x_i^*=(S_2-S_1a_i)/\Delta$, with minimum $S_2/\Delta$. To prove sufficiency, the <optimization Lagrangian> with multipliers $\lambda=S_2/\Delta$, $\mu=-S_1/\Delta$ is a sum of squares in $x_i-\lambda-\mu a_i$ plus a constant. Equivalently every feasible $x$ satisfies $\sum_i x_i^2-\sum_i(x_i^*)^2=\sum_i(x_i-x_i^*)^2$.
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