Some weird stuff happens in high-dimensions. Reference 1. High Dimensional Oranges Are Almost All Peel Consider an $n$-dimensional cube of side length 1 containing a smaller $n$-dimensional cube with side length $0.8$ (“pulp”) surrounded by a $0.1$-width border (“peel”). The volume of the pulp is $0.8^n$, which rapidly approaches 0 as $n$ increases: DimensionsPulp Volume ($0.8^n$) 10.800 20.640 30.512 50.328 100.107 200.012 500.000014 Another perspective: To randomly sample a point in...
