The "curse of dimensionality" means that in a hypercube the volume is increasingly concentrated in the corners as the number of dimensions increase.
In fact half the volume of a hypercube is closer to the centre than to the nearest vertex, with any number of dimensions.
The real curse is that the vast majority of the points of a unit hypercube of dimension $n$ are a distance less than $\frac{5}{n}$ from the outside of the hypercube, distances $\sqrt{\frac{n}{12}} \pm \frac12$ both from the centre and from the nearest vertex, and a distance $\sqrt{\frac{n}{6}} \pm 1$ from the vast majority of other points, which for large $n$ are narrow bands.