We could add constraints to make it meaningful. For example, restricting the profile to only an Lp norm superellipse [0], what is the exact norm value that res

We could add constraints to make it meaningful. For example, restricting the profile to only an Lp norm superellipse [0], what is the exact norm value that results in 1/2 volume? A 1-norm superellipse results in 1/3 volume (cone), 2-norm results in 2/3 volume (hemisphere), so it must be a norm between 1 and 2. A quick totally non-rigorous calculation on Desmos yields a power of 1.389857035315, you can see the shape in [1] [0] https://en.wikipedia.org/wiki/Lp_space https://en.wikipedia.org/wiki/Lp_space [1] https://www.desmos.com/calculator/x8y5evt0tc https://www.desmos.com/calculator/x8y5evt0tc