  Recently, Castagnoli et al. (2021) introduce the class of star-shaped risk
measures as a generalization of convex and coherent ones, proving that there is
a representation as the pointwise minimum of some family composed by convex
risk measures. Concomitantly, Jia et al. (2020) prove a similar representation
result for monetary risk measures, which are more general than star-shaped
ones. Then, there is a question on how both classes are connected. In this
letter, we provide an answer by casting light on the importance of the
acceptability of 0, which is linked to the property of normalization. We then
show that under mild conditions, a monetary risk measure is only a translation
away from star-shapedness.
