A spherical poylgon is a loop of geodesic segments on a 2-sphere,
each of whose side lengths is between 0 and the semicircumference
We prove that the side lengths of a spherical polygon satisfy the
spherical polygon inequalities. And conversely, given any lengths
which satisfies these, there exists a spherical polygon whose sides
have these lengths.
Given n unitary matrices Mk in
Mn = 1,
the n-gon inequalities are necessary conditions for the
simultaneous unitarizability of
M1, ... ,
are individually unitarizable and irreducible,
nk in [0, 1/2]
are defined by
cos 2 pi nk =
1/2 tr Mk,
then the spherical 3-gon inequalities imply that
This converse is false for n >3.
This problem has application to the construction of the moduli of
constant mean curvature genus zero surfaces with n Delaunay ends
(n-noids), where the asymptotic necksizes of the ends must satisfy the
See our paper below for proofs of the theorems and further details.