Theory (In-)Equivalence and conventionalism in f(R) gravity

f(R) Gravity is the most natural extension of General Relativity within Riemannian Geometry. Due to (inter alia) its potential capacity for a unified treatment of early and late-time cosmic expansion, it has enjoyed recent attention in astrophysics and cosmology. I critically examine three inter-related claims found in the pertinent physics literature, of general interest to the philosopher of science. 1. f(R) Gravity is equivalent to a particular Brans-Dicke Theory. 2. The spacetime geometry underpinning f(R) Gravity has substantial conventional elements. 3. f(R) Gravity is an instance of a theory in which the distinction between matter and spacetime is conventional. Whilst the first claim can be vindicated in precise terms, the remaining two claims, I submit, are unwarranted - at least for the reasons usually adduced. On different grounds, though, the case for conventionalism about spacetime geometry in f(R) Gravity (as well as General Relativity) turns out to be considerably stronger.