Holographic $c$-theorem and Born-Infeld gravity theories

The requirement of the existence of a holographic c-function for higher derivative theories is a very restrictive one, and hence most theories do not possess this property. Here, we show that, when some of the parameters are fixed, the $D\ge 3$ Born-Infeld gravity theories admit a holographic c-function. We work out the details of the $D=3$ theory with no free parameters, which is a nonminimal Born-Infeld type extension of new massive gravity. Moreover, we show that these theories generate an infinite number of higher derivative models admitting a c-function in a suitable expansion, and therefore they can be studied at any truncated order.