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This is particularly attributed to the observation that TV-penalties often favor piecewise constant minimizers with well-behaved jumpsets. On the downside, their intricate properties significantly complicate every aspect of their analysis, from the derivation of first-order optimality conditions to their discrete approximation and the choice of a suitable solution algorithm. In this paper, we investigate a general class of minimization problems with TV-regularization, comprising both continuous and discretized control spaces, from a convex geometry perspective. This leads to a variety of novel theoretical insights on minimization problems with total variation regularization as well as tools for their practical realization. First, by studying the extremal points of the respective total variation unit balls, we enable their efficient solution by geometry exploiting algorithms, e.g. fully-corrective generalized conditional gradient methods. We give a detailed account on the practical realization of such a method for piecewise constant finite element approximations of the control on triangulations of the spatial domain. 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