In problems of medical imaging and robotic vision, measurements of a convex set are sometimes obtained via the set's support function. The standard way in which the convex set is recovered from such data is to suppose that it is polygonal, model the errors as Normal random variables, and apply constrained maximum likelihood methods. Typically, the number of sides assumed of the polygon is equal to or a little less than the number of data points. However, from a statistical viewpoint, the number of sides should really be interpreted as a smoothing parameter and chosen to optimize some measure of performance. Additionally, if the true set is not a polygon, then a polygonal estimate can be aesthetically unsatisfactory. In this article we suggest periodic smoothing methods for estimating the convex set.