  The phase behavior of binary fluid mixtures of hard hyperspheres in four and
five dimensions is investigated. Spinodal instability is found by using a
recent and accurate prescription for the equation of state of the mixture that
requires the equation of state of the single component fluid as input. The role
played by the dimensionality on the possible metastability of the demixing
transition with respect to a fluid-solid transition is discussed. The binodal
curves in the pressure--chemical potential representation are seen to lie on a
common line, independent of the size ratio
