Generalization of Karmarkar's algorithm to convex homogeneous functions

Let φ be a convex homogeneous function of degree K > 0 defined over the positive points of a subspace W of Rⁿ, n ≥ 2. Assume φ(x) > 0 for some 0 < ε{lunate} W. Let V be the set of nonnegative nonzero x ε{lunate} W such that φ(x)^k converges to zero, for some sequence 0 < x ^k ε{lunate} W converging to x. We prove V is nonempty if and only if for every 0 < ε{lunate} W satisfying φ(d) > 0, there exists 0 < x_d ε{lunate} W such that e^Td = e^Tx_d and f(x_d) ≤ γf(d), where e is the vector of ones, f(x) = φ(x)/Πⁿ_i=1^xi^K/n)), and γ = 〈[K + 1)/K]^K exp(-1)〉^1/n. Karmarkar's algorithm proves the above for the special case where φ is linear.