  In this paper we present a new kind of semigroups called convex body
semigroups which are generated by convex bodies of R^k. They generalize to
arbitrary dimension the concept of proportionally modular numerical semigroup
of [7]. Several properties of these semigroups are proven. Affine convex body
semigroups obtained from circles and polygons of R^2 are characterized. The
algorithms for computing minimal system of generators of these semigroups are
given. We provide the implementation of some of them.
