Title: Slice, Dice, and Optimize: Measuring the Dimension of Neural Network Class Manifolds. Abstract: Deep neural network classifiers naturally partition input space into regions belonging to different classes. The geometry of these class manifolds (CMs) is widely studied and is intimately related to model performance; for example, the margin is defined via boundaries between these CMs. We present a simple technique to estimate the effective dimension of CMs as well as boundaries between multiple CMs, by computing their intersection with random affine subspaces of varying dimension. We provide a theory for the technique and verify that our theoretical predictions agree with measurements on real neural networks. Through extensive experiments, we leverage this method to show deep connections between the geometry of CMs, generalization, and robustness. In particular we investigate how CM dimension depends on 1) the dataset, 2) architecture, 3) random initialization, 4) stage of training, 5) class, 6) ensemble size, 7) label randomization, 8) training set size, and 9) model robustness to data corruption. Together a picture emerges that well-performing, robust models have higher dimensional CMs than worse performing models. Moreover, we offer a unique perspective on ensembling via intersections of CMs. Our core code is available on Github (link in the PDF abstract).