  We study the properties of unitary orbits of mixed spin states that are
characterized by Wigner functions lower bounded by a specified value. To this
end, we extend a characterization of the set of absolutely Wigner positive
states as a set of linear eigenvalue constraints, which together define a
polytope in the simplex of spin-j mixed states centred on the maximally mixed
state. The lower bound determines the relative size of such absolutely Wigner
bounded (AWB) polytopes and we study their geometric properties. In particular,
in each dimension a Hilbert-Schmidt ball representing a tight AWB sufficiency
criterion based on the purity is exactly determined, while another ball
representing AWB necessity is conjectured. Special attention is given to the
case where the polytope separates orbits containing only positive Wigner
functions from other orbits because of the use of Wigner negativity as a
witness of non-classicality of spin states. Comparisons are made to absolute
symmetric state separability and spherical Glauber-Sudarshan positivity, with
additional details given for low spin quantum numbers.
