The risk of financial positions is measured by the minimum amount of capital
to raise and invest in eligible portfolios of traded assets in order to meet a
prescribed acceptability constraint. We investigate nondegeneracy, finiteness
and continuity properties of these risk measures with respect to multiple
eligible assets. Our finiteness and continuity results highlight the interplay
between the acceptance set and the class of eligible portfolios. 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. Finally, we discuss applications to set-valued risk
measures, superhedging with shortfall risk, and optimal risk sharing
