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 *M*_{k} in
SU_{2},
with *M*_{1}...
*M*_{n} = 1,
the *n*-gon inequalities are necessary conditions for the
simultaneous unitarizability of
*M*_{1}, ... ,
*M*_{n}.

Conversely, if
*M*_{1},
*M*_{2},
*M*_{3} in
SL_{2}(C) with
*M*_{1}
*M*_{2}
*M*_{3}=1
are individually unitarizable and irreducible,
and
*n*_{k} in [0, 1/2]
are defined by
cos 2 pi n_{k} =
1/2 tr *M*_{k},
then the spherical 3-gon inequalities imply that
*M*_{1},
*M*_{2},
*M*_{3} 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.