On extremal points for some vectorial total variation seminorms

TL;DR

The paper investigates the extremal structure of generalized unit balls induced by vector-valued total variation seminorms $\operatorname{TV}_K$ and the related total deformation $\operatorname{TD}_K$ on bounded domains. By employing primal representations, measures on BV/BD spaces, and symmetry arguments for matrix norms, it provides precise extremal characterizations in the 1D vector- and scalar-valued multidimensional cases, shows additive norms reduce to scalar extrema, and demonstrates that highly isotropic norms generate richer families of extremals including multi-region piecewise constants. It also identifies two-region infinitesimally rigid extremals for $\operatorname{TD}_K$ and proves that the planar unit radial field (hedgehog) $u_H(x)=\frac{x}{|x|}$ is extremal for Frobenius TV, highlighting sharp contrasts between isotropic and anisotropic settings. These results inform sparse representer theorems, generalized conditional gradient methods, and continuum-mechanical models, where extremal structures guide efficient optimization and regularization strategies.

Abstract

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 are central to the understanding of sparse solutions and sparse optimization algorithms for variational problems posed among such functions. For 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 consist of piecewise constant functions with two regions. For definitions involving more involved matrix norms and in particular spectral norms, 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. For this case, we show piecewise infinitesimally rigid functions with two pieces to be extremal under mild assumptions. Finally, as an example which is not piecewise constant, we prove that unit radial vector fields are extremal for the Frobenius total variation in the plane.

Theorems & Definitions (34)

• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Theorem 3.3
• Remark 3.4
• Theorem 3.5
• proof
• Lemma 4.1
• proof