  Motivated by applications in robotics and computer vision, we study problems
related to spatial reasoning of a 3D environment using sublevel sets of
polynomials. These include: tightly containing a cloud of points (e.g.,
representing an obstacle) with convex or nearly-convex basic semialgebraic
sets, computation of Euclidean distances between two such sets, separation of
two convex basic semalgebraic sets that overlap, and tight containment of the
union of several basic semialgebraic sets with a single convex one. We use
algebraic techniques from sum of squares optimization that reduce all these
tasks to semidefinite programs of small size and present numerical experiments
in realistic scenarios.
