  In \cite{botero}, a top intersection product of toric b-divisors on a smooth
complete toric variety is defined. It is shown that a nef toric b-divisor
corresponds to a convex set and that its top intersection number equals the
volume of this convex set. The goal of this article is to extend this result
and define an intersection product of sufficiently positive toric b-classes of
arbitrary codimension. For this, we extend the polytope algebra of McMullen
(\cite{McM1}, \cite{McM2}, \cite{McM3}) to the so called \emph{convex-set
algebra} and we show that it embeds in the toric b-Chow group. In this way, the
convex-set algebra can be viewed as a ring for an intersection theory for
sufficiently positive toric b-classes. As an application, we show that some
Hodge type inequalities are satisfied for the convex-set algebra.
