Curse of Dimensionality and intuition
Yaroslav has this thought provoking article on the Curse of Dimensionality:
(…) consider a cube of width 1. As dimension increases, the volume stays the same. But (…) eventually almost all the mass is concentrated in the corners (meaning outside of the inscribed sphere).
The plot is especially shocking: at d=8, the sphere of diameter 1 inscribed in the unit cube has a negligible volume!
See
C. Zong, What is known about unit cubes, Bull.
Amer. Math. Soc. 42 no. 2 (2005) 181–211.
for many even more strange properties of the unit cubes.
Comment by Alexandre Borovik — 28/8/2006 @ 6:03