What shape are unit balls in $\mathbb R^d$ with the $L_p$ metric for $2 < p < \infty$?

Asked Apr 15 '23 at 01:16

Active Apr 15 '23 at 01:16

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Consider the below figure drawing some examples of unit balls for various $L_p$ metrics.

For $p=1$ and $p=\infty$ we get squares (hyperoctahedra and hypercubes in higher dimensions, respectively), and when $p=2$ we get a circle (spheres in higher dimensions).

For intermediary values of $p$, how would one describe what shape these balls are?

Would they be considered ellipses?

asked Apr 15 '23 at 01:16

pyridoxal_trigeminus

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Popular term: superellipse ($d=2$) and superellipsoid ($d > 2$). https://en.wikipedia.org/wiki/Superellipse . Also "Lamé curve" ($d=2$). – GEdgar Apr 15 '23 at 01:25
Ah, thank you @GEdgar - this is exactly what I was looking for and opens the door to learn more about these kinds of shapes. – pyridoxal_trigeminus Apr 15 '23 at 01:28
That article says the case $d=2, p=4$ is called a "squircle". But I must admit I have not heard this term elsewhere. – GEdgar Apr 15 '23 at 01:32
I noticed that. Apparently squircles generalize to spherical cubes, or "hypersphubes" (at least according to someone!). https://en.wikipedia.org/wiki/Squircle#p-norm_notation – pyridoxal_trigeminus Apr 15 '23 at 01:34
You might be interested in this post. – Ng Chung Tak Apr 15 '23 at 06:23

What shape are unit balls in $\mathbb R^d$ with the $L_p$ metric for $2 < p < \infty$?

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