# Some connections between measure partitions and polynomial partitioning

## Abdul Basit, Rutgers University

## April 28, 2015

In 1960, Grünbaum asked if any convex body in $\mathbb{R}^d$ could be
partitioned into $2^d$ parts of equal volume using
$d$-hyperplanes. While this question is still not completely resolved,
it has been generalized in quite interesting ways. In particular, the
following question is of interest.
What is the smallest dimension $d$, such that $j$ $d$-dimensional
measures can be partitioned into $2^k$ parts using $k$ hyperplanes.
The triples $(j, k, d)$ are called admissible triples.

More recently, Guth and Katz introduced equipartitions of point sets
using polynomials. These "polynomial partitions" played an important
part in their solution of the Erdős distinct distances problem in $\mathbb{R}^2$
and since then have been used to prove numerous bounds in Incidence
Geometry. Polynomial partitions themselves have been studied further,
and many interesting generalizations have been proved.

In this talk, I will show some connections between the problems
mentioned above. I will talk about some known results, and how these
can be generalized.