Kakeya Needle Sets
By using a trick of Pál, known as Pál joins (given two parallel lines, any unit line segment can be moved continuously from one to the other on a set of arbitrary small measure), a set in which a unit line segment can be rotated continuously through 180 degrees can be created from a Besicovitch set consisting of Perron trees.
In 1941, H. J. Van Alphen showed that there are arbitrary small Kakeya needle sets inside a circle with radius (arbitrary ). Simply connected Kakeya needle sets with smaller area than the deltoid were found in 1965. Melvin Bloom and I. J. Schoenberg independently presented Kakeya needle sets with areas approaching to, the Bloom-Schoenberg number. Schoenberg conjectured that this number is the lower bound for the area of simply connected Kakeya needle sets. However, in 1971, F. Cunningham showed that, given, there is a simply connected Kakeya needle set of area less than contained in a circle of radius 1.
Although there are Kakeya needle sets of arbitrarily small positive measure and Besicovich sets of measure 0, there are no Kakeya needle sets of measure 0.
Read more about this topic: Kakeya Set
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