Before revision:  We develop a new approach to dual representations of convex multi-asset risk measures which relies on a characterization of the structure of closedconvex acceptance sets. To avoid degenerate cases we need to ensure the existence of extensions of the underlying pricing functional which belong to the effective domain of the support function of the chosen acceptance set. We provide a characterization of when such extensions exist.
After revision:  We present a simple, alternative approach to the dual representation of convex risk measures by directly applying to the acceptance set the external characterization of closed, convex sets. We prove that risk measures are nondegenerate if and only if the pricing functional admits a positive extension which is a supporting functional for the underlying acceptance set, and provide a characterization of when such extensions exist.