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Every norm $\| \cdot \|_p$ defines a metric $d_p$. Using this metric, I want to calculate the circumference of the $\|1\|_p$-ball specific to that normed/metric space.

It was rather easy to calculate for the taxicab, euclidean and supremum metrics ($p=1,2,\infty$ respectively):

  • the $\|1\|_1$-ball has circumference $8$ (summing the $\| \cdot \|_1$-length of lines from vertex to vertex),
  • the $\|1\|_2$-ball has circumference $\tau = 2 \pi$ ("everyone knows that", but thinking about it, I can think of a central symmetry argument which allows relatively straightforward integration),
  • the $\|1\|_\infty$-ball has circumference $8$ (summing the $\| \cdot \|_\infty$-length of lines from vertex to vertex).

However the weird shape of lines

Is there a general formula for the circumference of the $\|1\|_p$-ball ? How does one find such a formula, ie, how does one calculate the circumference of the $\|1\|_p$-ball for any $p \in \; ]0, \infty[$ ? I expect some form of integral, but I'm not sure how to define it.

Side question: is there a standard way to define a (unique) measure (space) from a given metric (space)? (Ie, can I think of the Lebesgue measure as the measure generated by the euclidean metric, and expect the taxicab metric to generate another measure, or is that mistaken ?)

Tristan Duquesne
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    Seems related: [$\pi$ in arbitrary metric spaces](https://math.stackexchange.com/questions/254620/pi-in-arbitrary-metric-spaces?noredirect=1&lq=1), [measuring $\pi$ with alternate distance metrics](https://math.stackexchange.com/questions/2044223/measuring-pi-with-alternate-distance-metrics-p-norm?rq=1) – Calvin Khor Apr 27 '21 at 13:15
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    This answer should work for you. https://math.stackexchange.com/a/467454/773061 – Joshua Wang Apr 27 '21 at 13:37
  • Thanks a lot !! These offer nice context; and the specific integral pointed out seems to be exactly what I'm looking for; with the added bonus of giving me a way to calculate arc length with minor modifications. Now I need to understand how that was obtained... I'll try to get my hands on the jstor article, see if it helps. – Tristan Duquesne Apr 27 '21 at 17:33
  • @runway44 I'm pretty sure it's 8: the distance $|x - y|_1$ from point $(1, 0)$ to point $(0, 1)$ is 2, and you have 4 such segments on the $1$-circle of radius $1$. – Tristan Duquesne Nov 22 '21 at 11:39
  • Ah duh. ${}{}{}$ – runway44 Nov 22 '21 at 18:18

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