  This paper derives a discrete dual problem for a prototypical hybrid
high-order method for convex minimization problems. The discrete primal and
dual problem satisfy a weak convex duality that leads to a priori error
estimates with convergence rates under additional smoothness assumptions. This
duality holds for general polyhedral meshes and arbitrary polynomial degrees of
the discretization. A novel postprocessing is proposed and allows for
a~posteriori error estimates on regular triangulations into simplices using
primal-dual techniques. This motivates an adaptive mesh-refining algorithm,
which performs superiorly compared to uniform mesh refinements.
