Consider the closest pair of points using divide & conquer algorithm.

First, extend this algorithm to work on points in three dimensions. Note that the "crossover" pairs in the $3$D version of the problem, rather than sitting close to a line, sit close to a $2$d plane. That means there might be infinitely many of them, and we can't use the exact same approach we used in the $2$d version.

Second, construct a proof which uses your algorithm to calculate the closest pair of points IN $3$D from a set of points.

Finally, analyze the runtime complexity of your algorithm. You may not be able to use the Master Theorem to calculate the exact runtime, but if not use the Master Theorem to set upper and lower bounds on the runtime of your algorithm.