  The relaxation in the calculus of variation motivates the numerical analysis
of a class of degenerate convex minimization problems with non-strictly convex
energy densities with some convexity control and two-sided $p$-growth. The
minimizers may be non-unique in the primal variable but lead to a unique stress
$\sigma \in H(\operatorname{div},\Omega;\mathbb{M})$. Examples include the
p-Laplacian, an optimal design problem in topology optimization, and the
convexified double-well problem. The approximation by hybrid high-order methods
(HHO) utilizes a reconstruction of the gradients with piecewise Raviart-Thomas
or BDM finite elements without stabilization on a regular triangulation into
simplices. The application of this HHO method to the class of degenerate convex
minimization problems allows for a unique $H(\operatorname{div})$ conforming
stress approximation $\sigma_h$. The main results are a~priori and a posteriori
error estimates for the stress error $\sigma-\sigma_h$ in Lebesgue norms and a
computable lower energy bound. Numerical benchmarks display higher convergence
rates for higher polynomial degrees and include adaptive mesh-refining with the
first superlinear convergence rates of guaranteed lower energy bounds.
