Cyclic polygons with given edge lengths: Existence and uniqueness

Let a₁, ..., a_n be positive numbers satisfying the condition that each of the a_i’s is less than the sum of the rest of them; this condition is necessary for the a_i’s to be the edge lengths of a (closed) polygon. It is proved that then there exists a unique (up to an isometry) convex cyclic polygon with edge lengths a₁, ..., a_n. On the other hand, it is shown that, without the convexity condition, there is no uniqueness—even if the signs of all central angles and the winding number are fixed, in addition to the edge lengths.