# Effect of the curse of dimension on collision detection

I require a crude intersection analyzer or collision detector in about 15 dimensions. I am wondering if such a function is rendered difficult or impossible by the "curse of dimension". Collision detectors are widely used in 2 and 3 dimensions to determine if two shapes are in contact. A function that determines the intersection of two regions is one way to make a collision detector. The "curse of dimension" refers to difficulties encountered as the number of dimensions involved increases beyond about 5. For example, all points tend to be clustered near hyper-surfaces, and, statistically, all points behave like outliers. (See the Encyclopedia of Mathematics.) I haven't been able to find any literature on such collision detectors beyond 4 dimensions, and I would appreciate being pointed in the right (best?) direction.

If your objects fit into ellipsoids, then one option might be the Perram-Wertheim distance; in dimension $$n$$ you do need to invert an $$n \times n$$ matrix, which might be do-able ...