  We study families of depth measures defined by natural sets of axioms. We
show that any such depth measure is a constant factor approximation of Tukey
depth. We further investigate the dimensions of depth regions, showing that the
Cascade conjecture, introduced by Kalai for Tverberg depth, holds for all depth
measures which satisfy our most restrictive set of axioms, which includes Tukey
depth. Along the way, we introduce and study a new depth measure called
enclosing depth, which we believe to be of independent interest, and show its
relation to a constant-fraction Radon theorem on certain two-colored point
sets.
