This talk is devoted to the validity and the failure of the strong maximum principle for equations involving the k-th fractional truncated Laplacian or the k-th fractional eigenvalue, which are fully nonlinear integral operators whose nonlocality is somehow k-dimensional. We give in particular geometric characterizations of the sets of minima for nonnegative supersolutions. Based on joint works with I. Birindelli (Sapienza University), H. Ishii (Tsuda University) and D. Schiera (University of Lisbon).