Angles at which L1 ball and L2 ball touch

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Take L1 ball and L2 ball in $d$ dimensions. I'm interested in the angle $\alpha$ at the point of contact $p$.

Below is visualization for $d=2$

For more dimensions, there's more than one $\alpha$, so we can consider $\alpha$'s observed at $p$ when intersecting these surfaces with planes passing through 0 and $p$.

What is the smallest, largest $\alpha$ as a function of $d$? Is there a simpler expression when $d\to \infty$?

Some plane sections of L1 ball in 3D (notebook)

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