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 ?)
