Re:High-dimensional spheres are "spikey"
by altrego99 on Wednesday May 23, 2012 @12:55AM
(#4011605)
Thanks for the point - but your argument that 'from a surface almost every step leads you out' doesn't seem to be true in particular.
Imagine a sphere in n-dimensions, S = {Sum(x_i^2)<1}. Top of the sphere is p=(1,0,0,...,0). Now for any random direction r, if you take a very small step from p towards r, you have exactly 50% chance to be inside the sphere.
To be mathematically precise, for any r (unit vector) chosen at random, probability that there is e>0 such that p + e*r is within S is 1/2.
So half the time the steps take you out, half the time it takes you in - considering the step is small enough compared to the radius.
Imagine a sphere in n-dimensions, S = {Sum(x_i^2)<1}. Top of the sphere is p=(1,0,0,...,0). Now for any random direction r, if you take a very small step from p towards r, you have exactly 50% chance to be inside the sphere.
To be mathematically precise, for any r (unit vector) chosen at random, probability that there is e>0 such that p + e*r is within S is 1/2.
So half the time the steps take you out, half the time it takes you in - considering the step is small enough compared to the radius.