In this paper, we investigate the fine properties of functions under suitable geometric conditions on the jump set. Precisely, given an open set ω ⊂ Rn and given p > 1, we study the blow-up of functions u ε GSBV (ω), whose jump sets belong to an appropriate class Jp and whose approximate gradients are p-th power summable. In analogy with the theory of p-capacity in the context of Sobolev spaces, we prove that the blow-up of u converges up to a set of Hausdorff dimension less than or equal to n - p. Moreover, we are able to prove the following result which in the case of W1, p (ω) functions can be stated as follows: whenever u k strongly converges to u, then, up to subsequences, u k pointwise converges to u except on a set whose Hausdorff dimension is at most n - p {n-p}.
