  The entanglement wedge cross sections (EWCSs) are postulated as dual gravity
probes to certain measures for the entanglement of multiparty systems. We test
various proposed inequalities for EWCSs. As it turns out, contrary to
expectations, the EWCS is not clearly monogamous nor polygamous for tripartite
systems but the results depend on the details and dimensionality of the
geometry of the gravity solutions. We propose weaker monogamy relations for
dual entanglement measures, which lead to a new lower bound on EWCS. Our work
is based on a plethora of gravity backgrounds: pure anti de Sitter spaces, anti
de Sitter black branes, those induced by a stack of Dp-branes, and cigar
geometries in generic dimension.
