Additive manufacturing is a process by which a three dimensional object is created layer after layer, through selective deposition of material. It often requires the automated generation of auxiliary shapes, to temporarily support the object, to protect its surface, or to carve inner cavities and reduce material usage. In this context, we define a ``printable enclosure'' as a minimal volume enclosing a given shape and whose boundary can be printed at the smallest possible thickness while ensuring proper bonding between layers. Such an enclosure is well suited to serve as auxiliary structure for additive manufacturing: it is easy to print and require little material. In this paper, we demonstrate its use on three different applications: enclosing a print within protective walls that are close to the surface; generating large inner cavities whose walls are printable, and finally modeling support structures that provide a dense support to the downward facing surfaces while vanishing as quickly as possible below the supported object. We obtain the shape of an enclosure by considering constraints on its set of slices along horizontal planes. In practice, the set of slices is discrete and the constraints afford for an efficient sweep-like construction algorithm using morphological operations on the slices. We discuss the printability and optimality of the enclosures and their boundary walls.