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.