# Surface Area of a Superellipsoid

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 $$(x/a)^m +(y/b)^n+(z/c)^p=1; a \neq b\neq c; m \neq n\neq p$$

• There probably is no closed-form formula. – Matt F. Dec 29 '19 at 11:08
• @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? – Alexandre Eremenko Dec 29 '19 at 13:00
• @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. – M. Winter Dec 29 '19 at 14:59
• Some of the references I give in this 2 June 2007 sci.math post may be of interest. – Dave L Renfro Dec 30 '19 at 11:17

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
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?