Volumes of Lp unit balls

The unit ball in n dimensions under the Lp norm has volume

2^n frac{Gammaleft(1 + frac{1}{p}right)^n}{Gammaleft( 1 + frac{n}{p} right)}

I ran across this formula via A nice formula for the volume of an L_p ball. That post gives an even more general result that allows different values of p along each axis.

There have been several blog posts lately on the volume of balls in higher dimensions that correspond to the case p = 2. The formula above is valid for all p > 0.

Note that as p goes to ∞ the volume goes to 2n because the terms involving gamma functions go to 1. This is as we’d expect since the unit “ball” in the infinity norm is a cube, two units wide on each side.

Related post:

Means and inequalities

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5 comments on “Volumes of Lp unit balls
  1. For all p>0, not just p>=1? Cool!

  2. John says:

    Yes, p can be less than 1, leading to balls that are not convex.

  3. gappy says:

    Understanding the” shape” of a ball using an arbitrary norm-induced distance is central to many fields: approximation, theoretical CS, statistics, probability (esp. concentration of measure, with implication with random matrices and CLT), geometric and functional analysis. Most importantly, it’s an elementary topic (in terms of methods, not results) and revises our intuition, a bit like probability. The survey of Keith Ball is a great place to start (http://www.msri.org/publications/books/Book31/files/ball.pdf); the focus there is more on order of magnitude than on constant. Highly recommended.

    There are lots of beautiful results in this area: sections of balls (or of symmetric convex bodies), “widths” of balls (in the Gelfand or Kolmogorov sense), etc. They are waiting for an elementary exposition.

  4. Also note how the volume goes to zero while n->∞, due to faster growing of gamma function when compared to 2^n (also recall curse dimensionality).

  5. Asghar says:

    Dear John,
    thank you for your kindness to share this valuable information.
    I need to calculate the volume of a space which defined as follow:
    x1^p+….+xn^p<r 0<r<1.

    I would be grateful if you help me.
    thank you in advance

    Regards,
    Asghar

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