  We consider the set of extremal points of the generalized unit ball induced
by gradient total variation seminorms for vector-valued functions on bounded
Euclidean domains. These extremal points are central to the understanding of
sparse solutions and sparse optimization algorithms for variational
regularization problems posed among such functions. For not fully vectorial
cases in which either the domain or the target are one dimensional, or the sum
of the total variations of each component is used, we prove that these
extremals are fully characterized as in the scalar-valued case, that is, they
consist of piecewise constant functions with two regions. For definitions
involving more involved matrix norms and in particular spectral norms, which
are of interest in image processing, we produce families of examples to show
that the resulting set of extremal points is larger and includes piecewise
constant functions with more than two regions. We also consider the total
deformation induced by the symmetrized gradient, for which minimization with
linear constraints appears in problems of determination of limit loads in a
number of continuum mechanical models involving plasticity, bringing relevance
to the corresponding extremal points. For this case, we show piecewise
infinitesimally rigid functions with two pieces to be extremal under mild
assumptions. Finally, as an example of an extremal which is not piecewise
constant, we prove that unit radial vector fields are extremal for the
Frobenius total variation in the plane.
