This beautiful question was posed by the Japanese mathematician Sōichi Kakeya in 1917. It gained immediate prominence and, together with its higher-dimensional analogs, helped initiate a whole new field, today called geometric measure theory. To be precise, by “turning around” Kakeya had a continuous motion in mind that returns the needle to the original position with its ends reversed, like a Samurai whirling his pole. Any such motion takes place in a compact subset of the plane.
CITATION STYLE
Aigner, M., & Ziegler, G. M. (2018). The finite Kakeya problem. In Proofs from THE BOOK (pp. 247–251). Springer Berlin Heidelberg. https://doi.org/10.1007/978-3-662-57265-8_35
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