Adversarial attacks against deep neural networks are commonly constructed under
ℓp norm constraints, most often using
p=1,
p=2 or
p=∞, and potentially regularized for specific demands such as sparsity or smoothness. These choices are typically made without a systematic investigation of how the norm parameter
p influences the structural and perceptual properties of adversarial perturbations. In this work, we study how the choice of
p affects sparsity and smoothness of adversarial attacks generated under
ℓp norm constraints for values of
p∈[1,2]. To enable a quantitative analysis, we adopt two established sparsity measures from the literature and introduce three smoothness measures. In particular, we propose a general framework for deriving smoothness measures based on smoothing operations and additionally introduce a smoothness measure based on first-order Taylor approximations. Using these measures, we conduct a comprehensive empirical evaluation across multiple real-world image datasets and a diverse set of model architectures, including both convolutional and transformer-based networks. We show that the choice of
ℓ1 or
ℓ2 is suboptimal in most cases and the optimal
p value is dependent on the specific task. In our experiments, using
ℓp norms with
p∈[1.3,1.5] yields the best trade-off between sparse and smooth attacks. These findings highlight the importance of principled norm selection when designing and evaluating adversarial attacks.