What would the inequality $|x\pm y| \le |1\pm z|$ become in sphereical polar coordinates $r, \theta, \phi$, such that $(x,y,z) = (r \sin \theta \cos \phi, r \sin \theta \sin \phi, r \cos \theta)$?
To proceed, one may note that $|x\pm y| \le |1\pm z|$ represents a set of four inequalities $z \ge x+y-1; ~z\le -x+y+1;~z\le x-y+1;~z \ge -x-y-1.$ But how to translate these in terms of $r,\theta$, and $\phi$, such that $r\le 1$?