  The mean vertical heat transport $\langle wT \rangle$ in convection between
isothermal plates driven by uniform internal heating is investigated by means
of rigorous bounds. These are obtained as a function of the Rayleigh number $R$
by constructing feasible solutions to a convex variational problem, derived
using a formulation of the classical background method in terms of quadratic
auxiliary functions. When the fluid's temperature relative to the boundaries is
allowed to be positive or negative, numerical solution of the variational
problem shows that best previous bound $\langle wT \rangle \leq 1/2$ can only
be improved up to finite $R$. Indeed, we demonstrate analytically that $
\langle wT \rangle \leq 2^{-21/5} R^{1/5}$ and therefore prove that $\langle
wT\rangle< 1/2$ for $R < 65\,536$. However, if the minimum principle for
temperature is invoked, which asserts that internal temperature is at least as
large as the temperature of the isothermal boundaries, then numerically
optimised bounds are strictly smaller than $1/2$ until at least $R =3.4\times
10^{5}$. While the computational results suggest that the best bound on
$\langle wT\rangle$ approaches $1/2$ asymptotically from below as $R\rightarrow
\infty$, we prove that typical analytical constructions cannot be used to prove
this conjecture.
