Thanks very much Iosif - I will check that now.

I checked that RP_ - but found for $n=3$ the constant functions $f_A=f_B=0$ don't give a max (they give 5/9), instead I get a max (6/9) for example for $f_A(0)=1, f_A(1)=f_A(2)=0$, $f_B(0)=0, f_B(1)=f_B(2)=1$.

If one removes the $n^2$ in the denominator - this becomes an integer sequence, maybe it is in the OEIS........

Ah, okay, I guess I have not thought about the metric for the area A, or for the averaging: But then one can still pose this question for discrete case, area corresponding to N points, divide by N for average (as replied by Joseph below).

Yes, $\mathbb{R}^2$. General metric (not necessarily translation invariant). No specific metric in mind. The measure for distance is the metric.