On a non-homogeneous version of a problem of Firey

We investigate the uniqueness for the Monge-Ampere type equation det(u(ij) + delta(ij)u)(i,j=1)(n-1) = G(u), on Sn-1, (1) where u is the restriction of the support function on the sphere Sn-1, of a convex body that contains the origin in its interior and G : (0, infinity) -> (0, infinity) is a continuous function. The problem was initiated by Firey (Mathematika 21(1): 1-11, 1974) who, in the case G(theta) = theta(-1), asked if u equivalent to 1 is the unique solution to (1). Recently, Brendle et al. (Acta Mathe 219(1): 1-16, 2017) proved that if G(theta) = theta(-p), p > -n - 1, then u has to be constant, providing in particular a complete solution to Firey's problem. Our primary goal is to obtain uniqueness (or nearly uniqueness) results for (1) for a broader family of functions G. Our approach is very different than the techniques developed in Brendle et al. (2017).