Hyperspheres & the curse of dimensionality

I previously talked about the curse of dimensionality (more than 2 years ago) related to Machine Learning. Here I wanted to discuss it in more depth and dive into the mathematics of it. High dimensions might sound like Physics' string theory where our universe is made of more than 4 dimensions. This isn't what we are talking about here. The curse of dimensionality is related to what happens when a model deals with a data space with dimensions in the hundreds or thousands.