The Kakeya-Besicovitch problem was first proposed by S. Kakeya in 1917 and first solved by A.S. Besicovitch in 1928. A simpler proof was given later that year by O. Perron using what are now called "Perron trees."
Statement of the problem: Let AB be a unit segment in the plane. Move AB from its original position in such a way as to bring it back to its original position with its endpoints reversed, so that the final position is BA, and during this motion, AB should sweep out the least possible area.A simplistic first attempt might turn AB around A by 180 degrees and then to slide AB' along its line back to the final position BA.
Figure 1: Rotating the line AB about A 180 degrees. The line sweeps out an area equal to pi/2.
We can reduce the area by a factor of 2 by rotating around the midpoint:
Figure 2: Rotating the line AB about its midpoint results in an area of only pi/4.
Here are these first two attempts shown together:
Figure 3:
In 1917, the Japanese mathematician S. Kakeya considered the three-cusped hypocycloid. This is the curve described by a point on the rim of a circle of radius r that rolls without slipping within a fixed circle of radius R = 3r.
Figure 4: Three-cusped hypocycloid.
Kakeya conjectured that this was the curve that swept out the least possible area by taking a line segment and moving it in such a way so that it is always tangent to the curve. In moving along the curve, the line segment will end up with reverse orientation (try it on pencil and paper, or view an animation). It was a good guess, but he was wrong! The solution to the problem came from Besicovitch who, remarkably, showed that there were curves that could sweep out arbitrarily small areas. See [Besicovitch 1963] for details.
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