Curse of dimensionality when dimensions are fixed - dimensions

Question

I think there is a big misunderstanding in the Data Science community in respect to what exactly 'curse of high dimensionality' means. Please consider two examples:
1) I want to compare the distance between point A and point B in a 1000-dimensional and 1001-dimensional space. This is an example of curse of high dimensionality because there is a high chance that the distance will be higher in the 1001-dimensional space.
2) I want to compare the distance between point A and point B in a 1000-dimensional space, and a distance between point A and point C in the 1000-dimensional space. This is not a curse of high dimensionality because even though the dimensions are high, they are kept fixed.
Is the second statement correct? If the distance ratio between points A-B is twice higher than A-C in a 2-dimensional space, I would expect to see twice higher distance ratio in 1000 dimensional space of the same points. This means that the curse of high dimensionality only occurs when one tries to compare distances between different numbers of dimensions.

Answer 1

I think I have answered this question with a little test. Therefore, I am going to leave here in case it is going to be useful for somebody:
I did an experiment where I created a dummy data-set with 3 observations (A=1, B=2, C=4), calculated euclidean distance between points, and varied a number of features to see if the ratio of distances between the points start to differentiate as the features increase.
After 2 features:
0 1 2 ratio
0 0.00 1.41 4.24 3.00
1 0.00 1.41 2.83 2.00
2 0.00 2.83 4.24 1.50
After 100 features:
0 1 2 ratio
0 0.00 10.00 30.00 3.00
1 0.00 10.00 20.00 2.00
2 0.00 20.00 30.00 1.50
After 1000 features:
0 1 2 ratio
0 0.00 31.62 94.87 3.00
1 0.00 31.62 63.25 2.00
2 0.00 63.25 94.87 1.50
After 10000 features:
0 1 2 ratio
0 0.00 100.00 300.00 3.00
1 0.00 100.00 200.00 2.00
2 0.00 200.00 300.00 1.50
What does this means? Curse of high dimensionality does not occur when dimensions are fixed. It can be seen that the ratio distance between first closest point (1) and second closest point (2) remains constant when number of dimensions are increasing.
To put it in perspective, yes you do travel longer to points but that makes sense as your total data space increases with each added feature. However, the ratio of travel between points is kept the same and that's what matters.
To be honest, I do not see such a problem with the famous 'curse of high dimensionality' unless you are in the situation where you need to compare same points in variant n of dimensions.