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  1. The volume of an $n$-dimensional ball of radius $1$ is given by the classical formula $$V_n=\frac{\pi^{n/2}}{\Gamma(n/2+1)}.$$ For small values of $n$, we have $$V_1=2\qquad$$ $$V_2\approx 3.14$$ $$V_3\approx 4.18$$ $$V_4\approx 4.93$$ $$V_5\approx 5.26$$ $$V_6\approx 5.16$$ $$V_7\approx 4.72$$ It is not difficult to prove that $V_n$ assumes its maximal value when $n=5$.

    Question. Is there any non-analytic (i.e. geometric, probabilistic, combinatorial...) demonstration of this fact? What is so special about $n=5$?

  2. I also have a similar question concerning the $n$-dimensional volume $S_n$ ("surface area") of a unit $n$-sphere. Why is the maximum of $S_n$ attained at $n=7$ from a geometric point of view?

note: the question has also been asked on MathOverflow for those curious to other answers.

  • 25
    Are the volumes really comparable? "Physically", they have different units. What does it mean to say that the area of a disk is less than the volume of a ball? Perhaps there is a way to make this meaningful; I'd wager that would be a prerequisite to having a satisfactory answer to this question.2010-12-27
  • 20
    Well, for an $n$-dimensional ball of radius $R$ we can consider the ratio $$\frac{V_n(R)}{R^n}.$$ This is a "dimensionless" quantity.2010-12-27
  • 0
    If $X_i$ are i.i.d. gamma$(1/2,1)$ rv's, then the probability density function at $x=\pi$ of the sum $X_1 + \cdots + X_{n+2}$ is given by $e^{-\pi} V_n$.2010-12-27
  • 0
    I still don't see any reason that the ratio $V_n(R)/R^n$ is so special. I mean, it is the ratio of an n-sphere with radius 1 by an n-cube with side 1. Why should you compare those particular figures? Why not divide the volume of the n-cube into that of its (n-dimensional) circumcircle (http://en.wikipedia.org/wiki/Circumcircle). This would give $n^{n/2}2^{-n}V_n(R)/R^n$. Or its inscribed circle (http://en.wikipedia.org/wiki/Inscribed_circle), giving $2^{-n}V_n(R)/R^n$. Or compare the circle/cube which have equal surface area, giving $V_n(R)(2n/S_n(R))^{n/(n-1)}$.2010-12-28
  • 0
    ...I can see some geometric significance in those.2010-12-28
  • 1
    "A similar question concerning the n-dimensional volume $S_n$ ("surface area")..." - this should probably be rephrased, but I can't figure out how...2010-12-28
  • 5
    @George: I don't know that Andrey was suggesting that $V_n/R^n$ was particularly special -- just that it was one way to address Rahul's comment.2010-12-28
  • 2
    A useless answer would be that the maximum occurs for $V_5$ because $\pi$ is, what it is. It would be great to see a more explicit geometric connection than this though!2010-12-28
  • 1
    A less useless answer is: the paper linked at the end of this comment discusses monotonicity of hyperspherical areas and volumes, and tries to characterize where the max occurs. So, as per their argument, $V_5$ is a max may be viewed as a consequence of of radius $r=1$. Link: http://www.springerlink.com/content/g41072362835r517/2010-12-28

4 Answers 4