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I understand that little is known on the surface area of the unit ball in $\ell_p$, excepting the cases $p=1,2,\infty$, but can anything be said on the relation between the surface areas for $p$ and its conjugate $p'$? I'm particularly interested in small dimensions (up to five or so).

I seek invariants for tests for some code I have which estimates the surface area.

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  • $\begingroup$ Anything? Sure, we can say something any time, but it would be helpful if you get a bit more specific about the particular kind of "something" you are interested in. $\endgroup$
    – fedja
    Commented Apr 19, 2018 at 16:00
  • $\begingroup$ Something like "S(p) = f(S(p'))" using the obvious notation and for some function f, inequalities would also be useful. I've edited the question with my motivation. $\endgroup$
    – J.J. Green
    Commented Apr 19, 2018 at 16:10
  • $\begingroup$ The code specifically for $\ell^p$ balls, or for general (convex) shapes? In the latter case you'd be better off with creating some fancy surfaces whose areas you know exactly but in a non-trivial way. $\endgroup$
    – fedja
    Commented Apr 19, 2018 at 16:20
  • $\begingroup$ The code is specifically for ℓp balls, my tests up to now are against the easy cases, I'd like some against the hard cases, and since explicit value are not (do no seem to be) known, I seek invariants. $\endgroup$
    – J.J. Green
    Commented Apr 19, 2018 at 21:25

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