Title: Limitations of Piecewise Linearity for Efficient Robustness Certification. Abstract: Certified defenses against small-norm adversarial examples have received growing attention in recent years; though certified accuracies of state-of-the-art methods remain far below their non-robust counterparts, despite the fact that benchmark datasets have been shown to be well-separated at far larger radii than the literature generally attempts to certify. In this work, we offer insights that identify potential factors in this performance gap. Specifically, our analysis reveals that piecewise linearity imposes fundamental limitations on the tightness of leading certification techniques. These limitations are felt in practical terms as a greater need for capacity in models hoped to be certified efficiently. Moreover, this is _in addition_ to the capacity necessary to learn a robust boundary, studied in prior work. However, we argue that addressing the limitations of piecewise linearity through scaling up model capacity may give rise to potential difficulties---particularly regarding robust generalization---therefore, we conclude by suggesting that developing _smooth_ activation functions may be the way forward for advancing the performance of certified neural networks.