mean variance - Covariance of a GMV portfolio with any asset

Why is that the covariance of a global minimum variance (GMV) portfolio in the efficient frontier with any asset is always the same?

Answer

Here is the full math proof. Let g be the GMV portfolio and p be another asset.

We have:

$$ \begin{align*} Cov(x_g, x_p) &= E[{w_g}^T (x- \overline{x}) {(x- \overline{x})}^Tw_p]\\ &= {w_g}^TE[(x- \overline{x}) {(x- \overline{x})}^T]w_p\\ &= {w_g}^T\Sigma w_p \\ &= (\displaystyle\frac{{i}^T {\Sigma}^{-1}}{C})\Sigma w_p\\ &= \displaystyle\frac{{i}^Tw_p}{C}\\ &= \displaystyle\frac{1}{C} \end{align*} $$

where $C = 1^T {\Sigma}^{-1} 1 $