Three characterizations of reducedness of spherical convex polygons

Denote by $S^2$ the two-dimensional sphere. A spherical convex body on $S^2$ which does not properly contain a spherical convex body of the same spherical thickness is called a reduced body. We give three characterizations of reducedness of spherical convex odd-gons on $S^2$. Analogous characterizations hold true also in the Euclidean and hyperbolic planes.