  In this paper, we characterize the class of extremal points of the unit ball
of the Hessian-Schatten total variation (HTV) functional. The underlying
motivation for our work stems from a general representer theorem that
characterizes the solution set of regularized linear inverse problems in terms
of the extremal points of the regularization ball. Our analysis is mainly based
on studying the class of continuous and piecewise linear (CPWL) functions. In
particular, we show that in dimension $d=2$, CPWL functions are dense in the
unit ball of the HTV functional. Moreover, we prove that a CPWL function is
extremal if and only if its Hessian is minimally supported. For the converse,
we prove that the density result (which we have only proven for dimension $d =
2$) implies that the closure of the CPWL extreme points contains all extremal
points.
