Editing Kakeya set (section)

===Kakeya maximal function===
A modern way of approaching this problem is to consider a particular type of [[maximal function]], which we construct as follows: Denote '''S'''<sup>''n''−1</sup> ⊂ '''R'''<sup>''n''</sup> to be the unit sphere in ''n''-dimensional space. Define <math>T_{e}^{\delta}(a)</math> to be the cylinder of length 1, radius δ > 0, centered at the point ''a'' ∈ '''R'''<sup>''n''</sup>, and whose long side is parallel to the direction of the unit vector ''e'' ∈ '''S'''<sup>''n''−1</sup>. Then for a [[locally integrable]] function ''f'', we define the '''Kakeya maximal function''' of ''f'' to be

:<math> f_{*}^{\delta}(e)=\sup_{a\in\mathbf{R}^{n}}\frac{1}{m(T_{e}^{\delta}(a))}\int_{T_{e}^{\delta}(a)}|f(y)|dm(y),</math>

where ''m'' denotes the ''n''-dimensional [[Lebesgue measure]]. Notice that <math>f_{*}^{\delta}</math> is defined for vectors ''e'' in the sphere '''S'''<sup>''n''−1</sup>.

Then there is a conjecture for these functions that, if true, will imply the Kakeya set conjecture for higher dimensions:

:'''Kakeya maximal function conjecture''': For all ε > 0, there exists a constant ''C<sub>ε</sub>'' > 0 such that for any function ''f'' and all δ > 0, (see [[lp space]] for notation)
::<math> \left \|f_{*}^{\delta} \right \|_{L^n(\mathbf{S}^{n-1})} \leqslant C_{\epsilon} \delta^{-\epsilon}\|f\|_{L^n(\mathbf{R}^{n})}. </math>