Most adversarial attacks on 3D point clouds perturb a large number of points, causing widespread geometric changes and limiting their applicability in real-world scenarios. While recent works explore sparse attacks by modifying only a few points, such approaches often struggle to maintain effectiveness due to the limited influence of individual perturbations. In this paper, we propose SCP, a sparse and cooperative perturbation framework that selects and leverages a compact subset of points whose joint perturbations produce amplified adversarial effects. Specifically, SCP identifies the subset where the misclassification loss is locally convex with respect to their joint perturbations, which is determined by checking the positive-definiteness of the corresponding Hessian block. The selected subset is then optimized to generate high-impact adversarial examples with minimal modifications. Extensive experiments show that SCP achieves 100\% attack success rates, surpassing state-of-the-art sparse attacks, and delivers superior imperceptibility to dense attacks with far fewer modifications. Codes will be made public upon paper acceptance.