I became interested in Kakeya sets because they have the interesting property that a Kakeya set in a projective plane cannot be a subset of a blocking set, and with the exception of the full plane with two lines removed, Kakeya sets are the only sets with this property.

As far as state of the art in higher dimensions, not much seems known. The article http://onlinelibrary.wiley.com/doi/10.1002/jcd.21507/full, by Maarten De Boeck makes some strides in 3-space, and is an interesting paper.

I became interested in Kakeya sets because they have the interesting property that a Kakeya set in a projective plane cannot be a subset of a blocking set, and with the exception of the full plane with two lines removed, Kakeya sets are the only sets with this property.

As far as state of the art in higher dimensions, not much seems known. The article http://onlinelibrary.wiley.com/doi/10.1002/jcd.21507/full, by Maarten De Boeck makes some strides in 3-space, and is an interesting paper.

UPDATE: Shouldn't post so late in the day. I should have said that the affine part of a Kakeya set has the property of not being contained in a blocking set, and that they are almost the only sets not containing a line with that property.