A general notion of trimmed regions for empirical distributions in d-space is introduced. The regions are called weighted-mean trimmed regions. They are continuous in the data as well as in the trimming parameter. Further, these trimmed regions have many other attractive properties. In particular they are subadditive and monotone which makes it possible to construct multivariate measures of risk based on these regions. Special cases include the zonoid trimming and the ECH (expected convex hull) trimming. These regions can be exactly calculated for any dimension. Finally, the notion of weighted-mean trimmed regions extends to probability distributions in d-space, and a law of large numbers applies.