Author(s)

Rongchuan Tao¹, Yingzi Yang², Xiaoxiao Zou³, Zifan Dong⁴, Siran Chen⁵

¹School of Management, Wuhan University, Wuhan, China.
²City University of Hong Kong, Hong Kong, China.
³Pilgrim School, Los Angeles, USA.
⁴Weiyu High School, Shanghai, China.
⁵The SMIC Private School, Shanghai, China.

This research paper concentrates on the Kakeya problem. After the introduction of historical issue, we provide a thorough presentation of the results of Kakeya problem with some examples of the early solutions as well as the proof of the final outcome of this problem, the solution of which is known as Besicovitch Set. We give 3 different construction of Besicovitch set as well as the intuition of construction, which is related to iterated integral of 2-variable real function. We also give the Cunningham construction in which the area of a simply connected Kakeya set can also tend to 0. Furthermore, we generalize the process of generating a Kakeya set into a Kakeya dynamic. The definition of multiplicity enables us to estimate the area of a Kakeya set. In following discussion we provided a conjecture related to the solution in particular range. Finally, the derivation of the Kakeya problem is presented.

Kakeya Needle Problem, Besicovitch Set

Tao, R. , Yang, Y. , Zou, X. , Dong, Z. and Chen, S. (2019) The Kakeya Problem. Advances in Pure Mathematics, 9, 78-110. doi: 10.4236/apm.2019.92006.