I was reading Hilbert's proof of the Hilbert-Waring theorem in this survey paper, and came across the following statement in page 25, which i rephrased a little, regarding the center of mass of a set of vectors, in the real vector space $V$ of homogeneous polynomials of degree 2k in 5 variables:
Let $T$ be the set of vectors in $V$ given by the forms $L=$($\alpha_{1}x_{1}+ ... +\alpha_{5}x_{5})^{2k}$ with $\alpha_{i}\in$R and $\alpha_{1}^{2}+ ... +\alpha_{5}^{2}\leq1.$ Then the center of gravity of the mass distribution of unit density throughout $T$ and zero elsewhere is given by $\\$: $\displaystyle g = \left. \int_{R}(\alpha_{1}x_{1}+ ... +\alpha_{5}x_{5})^{2k}d\alpha_{1}...d\alpha_{5} \middle/ \int_{R}d\alpha_{1}...d\alpha_{5} \right.\\ $
where $R$ is the region in $R^{5}$ defined by $\alpha_{1}^{2}+ ... +\alpha_{5}^{2}\leq1$
I don't understand how this calculation in $R^{5}$ will give us the desired center of mass in the vector space $V$, any help would be appriciated!