What is the volume of the solid defined by $x^4+y^4+z^4=1$ enclosed in a box of unit length?

Question

I came across a post on the geometric interpretation of $x^3+y^3+z^3 = t^3$ asking if it was a higher analogue of the Pythagorean theorem.

It made me wonder: if we plot $x^2+y^2+z^2=1$, of course we get a sphere. But if we plot $x^4+y^4+z^4=1$, we get this solid instead:

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I know by a result of Elkies that this has infinitely many rational points $x,y,z$. What else is known about this solid?

1. For one, what is its name?
2. Also, if we snugly enclose it in a box of unit length, what is its volume?

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P.S. For the case $x^5+y^5+z^5 = 1$

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