Monotonicity of the volume of four balls in hyperbolic space

Consider a collection of possibly overlapping balls in E d. Suppose the balls are rearranged so that the distances between the centers of the balls all increase or stay the same. Kneser and Poulsen independently conjectured in the 1950’s that the volume of the union must either increase or stay the same. Csikós proved that this is true in Euclidean space under a continuous expansion, and Connelly and Bezdek proved it for the Euclidean plane (not necessarily a continuous expansion). It is still an open question for higher dimensions. The conjecture can also be stated for balls in hyperbolic space. Csikós proved that it is true under a continuous expansion in hyperbolic space, but it is still an open question for a non-continuous expansion in hyperbolic space. We will show that the conjecture holds for 4 balls in H 2 or H 3.