 Solution

Projection onto the constant polynomials requires the residual to be orthogonal to $1$. For the original inner product,For the weighted inner product the corresponding normal equation isHenceIt increases monotonically towards one. Although the weight vanishes at zero, it is positive elsewhere, and a nonzero continuous function is nonzero on an interval of positive weight. Thus the weighted expression really is an inner product and the constant minimizer is unique.

The normalized weight $(2n+1)x^{2n}/2$ has total mass one and assigns mass $r^{2n+1}$ to $[-r,r]$ for each $0<r<1$. Its mass therefore concentrates near $-1$ and $1$, where $x^2=1$. This explains why the weighted best constant tends to one rather than to the unweighted spatial average. It is endpoint concentration in weighted constant approximation.