A function f : S is linearly homogeneous if and only if epi f is

A function f : S → ℜ is linearly homogeneous if and only if epi f is a cone.
The following useful proposition show how quasiconcavity and homogeneity combine to produce full concavity. Quasiconcavity ensures convexity of the upper contour sets, while homogeneity of degree k ≤ 1 strengthens this to convexity of the hypograph (see remark 3.13).