  Any solid object can be decomposed into a collection of convex polytopes (in
short, convexes). When a small number of convexes are used, such a
decomposition can be thought of as a piece-wise approximation of the geometry.
This decomposition is fundamental in computer graphics, where it provides one
of the most common ways to approximate geometry, for example, in real-time
physics simulation. A convex object also has the property of being
simultaneously an explicit and implicit representation: one can interpret it
explicitly as a mesh derived by computing the vertices of a convex hull, or
implicitly as the collection of half-space constraints or support functions.
Their implicit representation makes them particularly well suited for neural
network training, as they abstract away from the topology of the geometry they
need to represent. However, at testing time, convexes can also generate
explicit representations -- polygonal meshes -- which can then be used in any
downstream application. We introduce a network architecture to represent a low
dimensional family of convexes. This family is automatically derived via an
auto-encoding process. We investigate the applications of this architecture
including automatic convex decomposition, image to 3D reconstruction, and
part-based shape retrieval.
