  We study a class of semialgebraic convex bodies called discotopes. These are
instances of zonoids, objects of interest in real algebraic geometry and random
geometry. We focus on the face structure and on the boundary hypersurface of
discotopes, highlighting interesting birational properties which may be
investigated using tools from algebraic geometry. When a discotope is the
Minkowski sum of two-dimensional discs, the Zariski closure of its set of
extreme points is an irreducible hypersurface. In this case, we provide an
upper bound for the degree of the hypersurface, drawing connections to the
theory of classical determinantal varieties.
