Euclidean-squared max-cut in low dimensions
The constant does tend to 1/2 as the dimension increases. In d dimensions, you can have d+1 points at distance one from each other, so the the sum of distance-squared is ${d+1 \choose 2}$ and the maximum cut is at most $(d+1)^2/4$, which is a $\frac 12 \cdot \frac {d+1}{d}$ fraction of the total weight