  The risk of extreme environmental events is of great importance for both the
authorities and the insurance industry. This paper concerns risk measures in a
spatial setting, in order to introduce the spatial features of damages stemming
from environmental events into the measure of the risk. We develop a new
concept of spatial risk measure, based on the spatially aggregated loss over
the region of interest, and propose an adapted set of axioms for these spatial
risk measures. These axioms quantify the sensitivity of the risk measure with
respect to the space and are especially linked to spatial diversification. The
proposed model for the cost underlying our definition of spatial risk measure
involves applying a damage function to the environmental variable considered.
We build and theoretically study concrete examples of spatial risk measures
based on the indicator function of max-stable processes exceeding a given
threshold. Some interpretations in terms of insurance are provided.
