A fixed two-dimensional projection of a three-dimensional Brownian
motion is almost surely neighborhood recurrent; is this simultaneously true of
all the two-dimensional projections with probability 1? Equivalently:
three-dimensional Brownian motion hits any infinite cylinder with probability
1; does it hit all cylinders? This papers shows that the answer is no. Brownian
motion in three dimensions avoids random cylinders and in fact avoids bodies of
revolution that grow almost as fast as cones.