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 SU₂, with M₁... M_n = 1, the n-gon inequalities are necessary conditions for the simultaneous unitarizability of M₁, ... , M_n.

Conversely, if M₁, M₂, M₃ in SL₂(C) with M₁ M₂ M₃=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 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.