In this article the notion of leverage of a design point when fitting a linear regression model is interpreted geometrically. In the case of least squares fitting, the leverage indicators based on the diagonal of the hat matrix are widely applied. By interpreting these hat matrix indicators geometrically, leverage can be generalized to groups of design points, as well as to other methods of fitting. The article introduces a leverage indicator that is appropriate forL1regression and discusses some aspects of this new diagnostic. It is shown that, in the case ofL1regression, the leverage indicators have a precise interpretation. They tell us about the breakdown and/or the exactness of fit. As an application, the article considers the maximal possible breakdown value ofL1regression and the choice of designs that achieve this maximum.