I calculate the area of the convex hulls of $n+1$ random points in a unit $n$-cube for $n$ = $3$ to $10$. (I use scipy.spatial.convexhull).
For each $n$, I generate $1000$ sets to get the average area and I find that the average area decreases fast with increasing $n$.
Is this result expected? I can’t wrap my head around why. (If not, I’ve probably misunderstood the scipy library.)
Since the area of a unit $n$-cube is $2n$, I kind of expected the area to increase with n in my experiment.
It is expected for the average area of the convex hulls of random points in a unit n-cube to decrease with increasing n.
As n increases, the number of points in the cube also increases, which makes it more likely that the points will be distributed more evenly across the cube. With more evenly distributed points, the convex hull will enclose a smaller area of the cube, leading to a smaller average area.
Another way to think about it is as the dimension of the cube increases, the volume of the cube increases exponentially while the surface area of the cube increases only linearly. So, as the dimension increases, the points are more likely to be closer to the interior of the cube, resulting in smaller convex hull area.
It is also important to note that the function you used from scipy library is returning the area of the convex hull and not the area of the cube, and the area of the convex hull is generally smaller than the area of the cube.