  In this paper we analyze in detail a few questions related to the theory of
functions with bounded $p$-Hessian-Schatten total variation, which are relevant
in connection with the theory of inverse problems and machine learning. We
prove an optimal density result, relative to the $p$-Hessian-Schatten total
variation, of continuous piecewise linear (CPWL) functions in any space
dimension $d$, using a construction based on a mesh whose local orientation is
adapted to the function to be approximated. We show that not all extremal
functions with respect to the $p$-Hessian-Schatten total variation are CPWL.
Finally, we prove existence of minimizers of certain relevant functionals
involving the $p$-Hessian-Schatten total variation in the critical dimension
$d=2$.
