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The shape described by $x^2 + y^2 +z^2 = 1$ is a sphere. But what do you call the shape described by $x^4 + y^4 +z^4 = 1$? Does it have a name?

Wyck
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    Yes, the $L^4$ unit sphere. – Igor Rivin Feb 20 '18 at 00:05
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    not one that caught on. Dirichlet came up with a method for finding the volume of such shapes, along with more complicated integrals. Has the gamma function in it, $8 \Gamma \left( \frac{5}{4} \right)^3 / \Gamma \left( \frac{7}{4} \right)$ But not the surface area – Will Jagy Feb 20 '18 at 00:06
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    The region satisfying $|x|^\sigma+|y|^\sigma+|z|^\sigma\leq 1$ is sometimes called a superball; see e.g. this Inventiones paper by J.A. Rush http://www.digizeitschriften.de/en/dms/img/?PID=GDZPPN002106957 and this one by Elkies, Odlyzko and Rush http://www.digizeitschriften.de/dms/img/?PID=GDZPPN002108941 . – j.c. Feb 20 '18 at 00:26
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    Also the quartic Fermat surface (if you're doing algebraic geometry or Diophantine equations). – Noam D. Elkies Feb 20 '18 at 00:56
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    Related is the notion of a superellipse and a superellipsoid (some lesser known references can be found here). – Dave L Renfro Feb 20 '18 at 12:54

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