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Source: Quant interview.

For any positive integer $n$, let $A_n$ denote the surface area of the unit ball in $R^n$, and let $V^n$ denote the volume of the unit ball in $R^n$. Let $i$ be the positive integer such that $A_i > A_k$ for all $k \neq i$. Similarly let $j$ be the positive integer such that $V_j > V_k$ for all $k \neq j$.

What is $i-j$?

Attempt: Given that $V_n = \frac{A_n}{n}$ then $\Delta A_n = n \Delta V_n + V_n$ which shows that $i-j > 0$. I cannot find a better lower bound or upper bound (without deriving the closed form and checking numerically).

Mattiatore
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  • Classic nonsense question for a quant interview! – SBK Aug 30 '22 at 20:57
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    Not two weeks ago someone asked the same question, also from an interview. Don't recall if it was marked a duplicate of https://math.stackexchange.com/questions/15656/volumes-of-n-balls-what-is-so-special-about-n-5 but it perhaps could have been. – Andrew D. Hwang Aug 30 '22 at 21:03
  • Is linear algebra an appropriate tag? – ShawSa Aug 30 '22 at 21:58
  • Here it is: https://math.stackexchange.com/questions/4510048/relationship-between-volume-of-n-sphere-and-surface-area-of-n-1-sphere – Andrew D. Hwang Sep 07 '22 at 16:57

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