I was trying to calculate the side of an equilateral triangle with the vertices on the unit sphere of $l_p^2$, when $1<p<\infty$. When I say "equilateral", I mean with respect to the distance in $l_p^2$. With the exception of $p=2$, where is a trivial problem of Euclidean geometry, I am having problems finding the coordinates of the vertices, as I get some equations I don't seem to be able to solve.
What I tried was fixing one vertex at $(0,1)$ and the other two having coordinates $(a,b)$ and $(-a,b)$. I am unable to solve for $a$ and $b$.
- Any ideas on how I solve this, or perhaps a different approach?
- Are there any known results for the regular $n$-gon with vertices on the unit sphere of $l_p^n$. Again, I can calculate this pretty easily for $p=2$.
Edit: In view of Christian Blatter below, I will change the question to finding the largest $\lambda$ such that there exist $3$ points on the unit sphere of $l_p^2$ such that the distance between any $2$ is equal to $\lambda$. Similarly for $l_p^n$. I think such largest distance should exist by a simple compactness argument.