Abstract: Data-free Class-incremental Learning (CIL) is a challenging problem because rehearsing data from previous phases is strictly prohibited, causing catastrophic forgetting of Deep Neural Networks (DNNs). In this paper, we present \emph{iVoro}, a novel framework derived from computational geometry. We found Voronoi Diagram (VD), a classical model for space subdivision, is especially powerful for solving the CIL problem, because VD itself can be constructed favorably in an incremental manner -- the newly added sites (classes) will only affect the proximate classes, making the non-contiguous classes hardly forgettable. Furthermore, we bridge DNN and VD using Power Diagram Reduction, and show that the VD structure can be progressively refined along the phases using a divide-and-conquer algorithm. Moreover, our VD construction is not restricted to the deep feature space, but is also applicable to multiple intermediate feature spaces, promoting VD to be multilayer VD that efficiently captures multi-grained features from DNN. Importantly, \emph{iVoro} is also capable of handling uncertainty-aware test-time Voronoi cell assignment and has exhibited high correlations between geometric uncertainty and predictive accuracy (up to ${\sim}0.9$). Putting everything together, \emph{iVoro} achieves up to $25.26\%$, $37.09\%$, and $33.21\%$ improvements on CIFAR-100, TinyImageNet, and ImageNet-Subset, respectively, compared to the state-of-the-art non-exemplar CIL approaches. In conclusion, \emph{iVoro} enables highly accurate, privacy-preserving, and geometrically interpretable CIL that is particularly useful when cross-phase data sharing is forbidden, e.g. in medical applications.