By appealing to the Poincare-Hopf Theorem on topological invariants, we introduce
a global classification scheme for homogeneous, convex bodies based on the number
and type of their equilibria. We show that beyond trivially empty classes all other
classes are non-empty in the case of three-dimensional bodies; in particular we prove
the existence of a body with just one stable and one unstable equilibrium. In the
case of two-dimensional bodies the situation is radically different: the class with
one stable and one unstable equilibrium is empty (Domokos, Papadopoulos, Ruina, J.
Elasticity 36 [1994], 59-66). We also show that the latter result is equivalent to
the classical Four-Vertex Theorem in differential geometry. We illustrate the introduced
equivalence classes by various types of dice and statistical experimental results
concerning pebbles on the seacoast.