Euclidean distance (cosine) between two random positive unit vectors in high dimensional space I found out that the largest possible euclidean distance (which is the cosine) between two random positive unit vectors decreases as the dimension of vector increases and approximates 0.71. This was done by a simulation where I randomly sample unit vectors and compute the maximal pair-wise Euclidean distance. I found the value 0.71 intriguing. It's as if the maximal angle between all vectors is 45 degrees (cosine(45) ≈ 0.71), which is half of 90 degrees related to the constraint of having only positive elements.  
Why is that? I sense there is an intuitive explanation to this which can be generalized to other cases, e.g. without the positive vector constraints.
You can use the following code to reproduce the simulation.
import numpy as np
from sklearn.preprocessing import Normalizer
from sklearn.metrics.pairwise import cosine_similarity, euclidean_distances
matrices = [Normalizer(norm = 'l2').fit_transform(np.random.rand(1000, N)) for N in range(2,10000, 100)]
res = [euclidean_distances(matrix, matrix).max() for matrix in matrices]
plt.plot(range(len(res)),res)

plot of the max euclidean distance with different dimension

Take two random vectors. As $N\to\infty$, it becomes increasingly unlikely that they are (almost) parallel or (almost) orthogonal. In fact I think it should be straightforward to prove that the expected value of the cosine similarity is equal to $\cos(45^\circ)\approx 0.7$ for all $N\ge 3$, but the standard deviation goes to $0$ as $N\to\infty$. (solving the integrals may be hard however). The reason you do not see values $>0.7$ is because you do not sample enough points. Due to curse of dimensionality you will have to sample roughly $2^N$ many points in $N$ dimensions. 
