This is a homework question and I have been attempting it for a while. The problem is from the book "Real Mathematical Analysis" by Charles Pugh, Chapter 6 #4.
The question is asking for a generalization of a lemma in 2-dim to n-dim. The content of the lemma is stated in the title (n=2) and the "insight" was that a square $S$ contains a disk $\Delta$ such that $m\Delta / mS > 1/2$, where $m$ is the Lebesgue outer measure.
I am thinking of a corresponding inequality in n-dim, and I feel I should try proving $m\Delta / mS > 1/n$, where $S$ is the n-dim cube ($(-R, R)^n$) and $\Delta$ should be the n-ball with radius $R$.
I looked up the volume formula for n-ball, and think I should try showing the two inequalities (if my guess is right):
$n = 2k: V_{2k}(R) / (2R)^{2k} = (\frac{\pi}{4})^k \frac{1}{k!} > \frac{1}{2k}$
$n = 2k+1: V_{2k+1}(R) / (2R)^{2k+1} = \frac{\pi ^k k!}{(2k+1)!} > \frac{1}{2k+1}$
For the $n=2k$ case, I tried grouping the LHS as $\Pi_{i=1}^{k} (\frac{1}{i} \frac{\pi}{4})$ but found that $\Pi_{i=2}^{k} (\frac{1}{i} \frac{\pi}{4}) < \frac{1}{k}$, though when $i=1$ the term is greater than 1/2. I am stuck and do not know where to start, and any help will be greatly appreciated.