  This paper presents several new results related to the Kakeya problem. First,
we establish a geometric inequality which says that collections of
direction-separated tubes (thin neighborhoods of line segments that point in
different directions) cannot cluster inside thin neighborhoods of low degree
algebraic varieties. We use this geometric inequality to obtain a new family of
multilinear Kakeya estimates for direction-separated tubes. Using the linear /
multilinear theory of Bourgain and Guth, these multilinear Kakeya estimates are
converted into Kakeya maximal function estimates. Specifically, we obtain a
Kakeya maximal function estimate in $\mathbb{R}^n$ at dimension $d(n) =
(2-\sqrt{2})n + c(n)$ for some $c(n)>0$. Our bounds are new in all dimensions
except $n=2,3,4,$ and $6$.
