No-go results and promising directions in the space of causal diamonds

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The space of causal diamonds recently brought to attention by de Boer et al. and Czech et al. provides an organizing principle for the dependence of entanglement entropy in conformal field theories on the spatial subregion considered. I will show that the inclusion relation of causal diamonds does not give rise to a consistent notion of a causal structure and thus does not provide an alternate metric on this space. I will also show that the entanglement entropy of ball shaped regions is not enough to reconstruct the areas of higher dimensional bulk surfaces in a static geometry. Instead, I will provide a new direction for the reconstruction of the areas of bulk surfaces from boundary data in holography, using the shape derivatives of the entanglement entropy of nearly ball shaped regions.