I am currently working through a set of notes I found on the internet at: http://math.msu.edu/~charlesb/Notes/DuoChapter2.pdf
I am up to page 8, and the Hardy-Littlewood maximal function for balls has just been introduced. Then it says that we can also define maximal functions over cubes centred at $x$. Then there is the phrase: "Furthermore, since the $n$-dimensional volumes of the unit cube and unit ball are equal up to a multiplicative constant depending only on $n$, it is immediate that $Mf$ and $M'f$ are comparable in the sense that $c_nM'f(x)\leq Mf(x)\leq C_nM'f(x)$ for constants $c_n$ and $C_N$ only depending on $n$."
It may be immediate to the author but it is not at all to me! I cannot understand why this is true. Does anyone have a proof? And is there a formula for $c_n$ and $C_n$?
All I can think of is maybe it's possible to come up with some sort of comparison between the size of a ball and the size of a cube both using the same $r$, but then the integrals may not be equal in order to compare the entire maximal funnction...