  This paper introduces a novel approach for learning polynomial
representations of physical objects. Given a point cloud data set associated
with a physical object, we solve a one-class classification problem to bound
the data points by a polynomial sublevel set while harnessing Sum-of-Squares
(SOS) programming to enforce prior shape knowledge constraints. By representing
objects as polynomial sublevel sets we further show it is possible to construct
a secondary SOS program to certify whether objects are packed correctly, that
is object boundaries do not overlap and are inside some container set. While
not employing reinforcement learning (RL) in this work, our proposed secondary
SOS program does provide a potential surrogate reward function for RL
algorithms, autonomously rewarding agents that propose object rotations and
translations that correctly pack objects within a given container set.
