  We explore the notion of $c$-functions in renormalization group flows between
theories in different spacetime dimensions. We discuss functions connecting
central charges of the UV and IR fixed point theories on the one hand, and
functions which are monotonic along the flow on the other. First, using the
geometric properties of the holographic dual RG flows across dimensions and the
constraints from the null energy condition, we construct a monotonic
holographic $c$-function and thereby establish a holographic $c$-theorem across
dimensions. Second, we use entanglement entropies for two different types of
entangling regions in a field theory along the RG flow across dimensions to
construct candidate $c$-functions which satisfy one of the two criteria but not
both. In due process we also discuss an interesting connection between corner
contributions to the entanglement entropy and the topology of the compact
internal space. As concrete examples for both approaches, we holographically
study twisted compactifications of 4d $\mathcal N=4$ SYM and compactifications
of 6d $\mathcal N=(2,0)$ theories.
