A spherical poylgon is a loop of geodesic segments on a 2-sphere,
each of whose side lengths is between 0 and the semicircumference
inclusively.
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
SU2,
with M1...
Mn = 1,
the n-gon inequalities are necessary conditions for the
simultaneous unitarizability of
M1, ... ,
Mn.
Conversely, if
M1,
M2,
M3 in
SL2(C) with
M1
M2
M3=1
are individually unitarizable and irreducible,
and
nk in [0, 1/2]
are defined by
cos 2 pi nk =
1/2 tr Mk,
then the spherical 3-gon inequalities imply that
M1,
M2,
M3 are
simultaneously unitarizable.
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
n-gon inequalities.
See our paper below for proofs of the theorems and further details.