A Discussion of a Generalization of the Multilinear Kakeya Theorem

The Multilinear Kakeya Inequality is a geometric estimate on the overlap of tubes in Euclidean space useful to the understanding of high order interactions between wave packets in harmonic analysis. We will discuss the evolution of the problem of achieving this estimate from the linear Kakeya problem and the combinatorial heuristics of the joints problem. Finally, we will discuss a slight modification that replaces the tubes with neighborhoods of Lipschitz graphs.