  We study the space of generalized translation invariant valuations on a
finite-dimensional vector space and construct a partial convolution which
extends the convolution of smooth translation invariant valuations. Our main
theorem is that McMullen's polytope algebra is a subalgebra of the (partial)
convolution algebra of generalized translation invariant valuations. More
precisely, we show that the polytope algebra embeds injectively into the space
of generalized translation invariant valuations and that for polytopes in
general position, the convolution is defined and corresponds to the product in
the polytope algebra.
