A simple proof of the nonconcavifiability of functions with linear not-all-parallel contour sets

Consider a real-valued function that, on a convex subset of a real vector space, is continuous on line segments and has convex contour sets. Inspired by a compelling intuitive argument due to Aumann (1975), we provide a simple proof that no strictly increasing transformation of such a function can be concave unless all contour sets are parallel, i.e., unless for every pair of contour sets, either their affine hulls are disjoint or one of their affine hulls contains the other.