Approximation of Vorob'ev expectation for random closed sets

Random sets appear in many applications, in particular in image analysis. The issue of a "mean shape" often arises since there is no canonical definition. In this paper, we propose a consistent and ready to use estimator for the Vorob'ev expectation of a random set $X$. It is a kind of mean closely linked to quantile-like quantities and built from independent copies of $X$ with spatial discretization. The convergence is established through the Strong Law of Large Numbers of Kovyazin. The control of discretization errors is handled with a mild regularity assumption on the boundary of $X$: a not too large 'box counting' dimension. Some examples, including Boolean models, are studied.