TITLE:
The Kakeya Problem
AUTHORS:
Rongchuan Tao, Yingzi Yang, Xiaoxiao Zou, Zifan Dong, Siran Chen
KEYWORDS:
Kakeya Needle Problem, Besicovitch Set
JOURNAL NAME:
Advances in Pure Mathematics,
Vol.9 No.2,
February
20,
2019
ABSTRACT: 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.