We know the surface area of a supersphere (x^n+y^n+z^n=1) can be easily obtained using the gamma function (here is the discussion). But what happens when we consider a superellipsoid ($x^m +y^n+z^p=1; m \neq n\neq p $), or more generally for the implicit function $(x/a)^m +(y/b)^n+(z/c)^p=1; a \neq b\neq c; m \neq n\neq p $
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1$\begingroup$ There probably is no closed-form formula. $\endgroup$– user44143Commented Dec 29, 2019 at 11:08
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1$\begingroup$ @Matt F.: Depends of what is meant by "closed-form" formula. Any formula will contain some integral. But Gamma function (for the sphere) is also an integral, is not it? $\endgroup$– Alexandre EremenkoCommented Dec 29, 2019 at 13:00
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1$\begingroup$ @AlexandreEremenko I would argue that the Gamma function in the formulas for spheres is just there to give a common expression for both, odd and even dimensions. You do not need it for, say, even dimensions. Here you can give a closed form using only factorials. $\endgroup$– M. WinterCommented Dec 29, 2019 at 14:59
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$\begingroup$ Some of the references I give in this 2 June 2007 sci.math post may be of interest. $\endgroup$– Dave L RenfroCommented Dec 30, 2019 at 11:17
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The surface area of an $n$-dimensional ellipsoid is expressed in terms of hyperelliptic integrals, see SURFACE AREA AND CAPACITY OF ELLIPSOIDS IN n DIMENSIONS by Garry J. Tee
In dimension 3, they are elliptic integrals, and the result is due to Legendre. The paper also mentions approximate formulas.
For $n\neq m\neq p$ one can write the integrals but they have no standard name: Surface area of an $\ell_p$ unit ball?
answered Dec 29, 2019 at 12:58
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$\begingroup$ I have tried to solve with the article you have suggested, can you find out how to employ the integral for different exponents? $\endgroup$ Commented Jan 2, 2020 at 20:13