Abstract: In the past few years, new families of entropy inequalities have been discovered. In the context of holography, they describe an important class of quantum states, those whose entanglement entropies can be computed by minimal cuts through some bulk structure either a holographic spacetime or a random tensor network. Among these inequalities two stand out: the so-called toric and projective inequalities. The entropies involved in these inequalities turn out to organize as tessellations of the torus and the real projective plane. In this talk, I will explain work towards the understanding of the continuum limit of these inequalities. Interestingly, the physical meaning in the continuum limit is related to the entropy of two-sided black holes and differential entropy in kinematic space.