GPS calculations require finding a sphere externally tangent to four given spheres, an Apollonian problem in $\mathbb{R}^3$. The center of that fifth sphere is one of the $16$ possible solutions to the Apollonian problem, only two of which touch the given four from outside. These equations are generally solved via resultants.

I was wondering what would be the GPS calculations under, say, the $L^3$ norm, which measures the length of a vector $x=(x_1,x_2,x_3)$ as $$ \|x\|_3 = \sqrt[3]{|x_1|^3 + |x_2|^3 + |x_3|^3} \;. $$ And in particular:

Q. How many solutions are there to the Apollonian problem under the $L^p$ norm: Given $d{+}1$ $L^p$-spheres in dimension $d$, find another sphere tangent to those $d{+}1$ spheres. How many externally tangent solutions?

          Three circles under the $L^3$ norm, and a 4th externally tangent to the 1st three.


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