# Ratio of volumes of $n$-dimensional unit balls

Prove that $$\left( \frac{nS_n}{S_{n-1}}\right)^{1/n} \le 2$$ where $S_i$ is the volume of the $i$th dimensional unit ball and $n\ge 2$.

I think we can use the fact that an $n$-dimensional unit ball is contained in a hypercube whose edge measures $2$ units, and the ball contains a hypercube whose edge measures $\sqrt{2}$ units. So we have $\sqrt{2}^n\le S_n \le 2^n$ and thus $$\left( \frac{nS_n}{S_{n-1}}\right)^{1/n} \le \left( \frac{n2^n}{\sqrt{2}^{n-1}} \right)^{1/n} = n^{1/n}2^{\frac{n+1}{2n}} = n^{1/n}\sqrt{2}\sqrt{2}^{1/n}.$$

So if we take $n^{1/n} \le 2$ and $\sqrt{2}^{1/n}\le \sqrt{2}^{1/2}$, then an upper bound to my problem is $2^{1+3/4}$ which isn't good enough.

Do we need another approach or do we get better bounds on $n^{1/n}$?

• Since $3^{1 / 3} \sqrt(2) \sqrt{2}^{1 / n} > 2$, no estimate on $n^{1 / n}$ will work. – Travis Willse Jan 26 '18 at 12:55
• Your argument is very nice, but it only works if $n^{\frac{1}{n}}2^{\frac{n+1}{2n}} \le 2 \Leftrightarrow n \le 2^{\frac{n-1}{2}}$, which is true for all $n \ge 7$, since the function on the right is exponential. So if you can find a better bound for lower $n$'s it's fine, but it might require an explicit formula?? – 57Jimmy Jan 26 '18 at 12:55

## 2 Answers

It is enough to show that $S_n\leq 2\,S_{n-1}$, which is trivial, since

$$\{(x_1,\ldots,x_{n-1}):x_1^2+\ldots+x_{n-1}^2\leq 1\}\times[-1,1]\supset \{(x_1,\ldots,x_{n}):x_1^2+\ldots+x_{n}^2\leq 1\}.$$ Over $n\geq 2$, $(2n)^{1/n}$ attains its maximum, $2$, at $n=2$.

Since the volume of a $n$-dimensional unit ball is (see volume of an $n$- ball)$$S_n = \frac{π^{\frac{n}{2}}}{Γ\left( n + \frac{1}{2} \right)},$$ then\begin{align*} \left( \frac{nS_n}{S_{n - 1}} \right)^{\frac{1}{n}} \leqslant 2 &\Longleftrightarrow \frac{n \sqrt{π} \cdot Γ\left( n - \frac{1}{2} \right)}{Γ\left( n + \frac{1}{2} \right)} \leqslant 2^n\\ &\Longleftrightarrow \frac{n \sqrt{π}}{n - \frac{1}{2}} \leqslant 2^n. \end{align*} Because$$2^n \geqslant 4 > 2 \sqrt{π} \geqslant \frac{n \sqrt{π}}{n - \frac{1}{2}},$$ then $\displaystyle \left( \frac{nS_n}{S_{n - 1}} \right)^{\frac{1}{n}} \leqslant 2$.

• This looks like "cheating". I don't think you're supposed to assume that formula to begin with. – Deepak Jan 26 '18 at 14:02
• @Deepak Nowhere in the question prohibits use of explicit formulae for $S_n$. – Saad Jan 26 '18 at 14:07
• The use of the gamma function is fine. I was trying to find a more intuitive explanation but I came across this result in a paper where they casually slipped this in. They could have used the gamma function. – Picasso Jan 26 '18 at 14:15