  In this paper, we have investigated phase transitions in arbitrary spacetime
dimensions for Born-Infeld AdS black holes. The phase transition points are
characterised from the divergence of heat capacity of the black hole. Two well
established techniques, namely, the Ehrenfest scheme and the Ruppeiner state
space geometry approach are used to identify the order of phase transition the
black hole undergoes. It is observed that the results obtained from these two
methods agree with each other. Our analysis reveals that the phase transition
is of second order. It is also observed from the variation of the heat capacity
with entropy that the small unstable black hole phase becomes more and more
stable with increase in the spacetime dimensions. We speculate that this
dependence of the stability of the black hole on the spacetime dimension can
put an upper limit to the dimension of spacetime from the physical condition of
the improbability of the formation of a small stable black hole. We have also
derived a Smarr relation in D-spacetime dimensions using scaling arguments and
first law of black hole thermodynamics which includes the cosmological constant
and the Born-Infeld parameters as thermodynamic variables.
