# Some Geometric Partitioning Questions, Answers, Algorithms

## William Steiger, Rutgers University

## May 5, 2015

It is known that for a given convex body $K$ in $\mathbb{R}^3$, there are three
planes whose octants each cut off the same volume in $K$. In fact
"volume" can be replaced by a "nice" measure $m$. In addition for
dimension $d>4$, the analogous statement is in general, false.
The situation when $d=4$ is tantalizingly open.

This example highlights some main concerns of *geometric
partitioning,* namely what kinds of partitions of geometric objects
always exist and what kinds need not exist. Another aspect of this
topic is an algorithmic one that arises when the measure m is counting
measure w.r.t. a given set of n points. Here a concern is to learn
the complexity of the partitioning task, as well as to create good
algorithms for it. I will discuss these ideas through a set of
examples in which there are some known results, and also some new
ones.