We discuss recent progress with Victor Lie on a type of inverse Kakeya problem in two dimensions. If the Cordoba-Kakeya estimate is sharp, can we say anything meaningful about the set of bad points? What do the level-sets of the rectangles look like? This is related to the family of problems including Bourgain's sum-product theorem, Katz-Tao ring conjecture, and Furstenburg's generalization of the Kakeya problem.
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