Title: Islands of Confidence: Robust Neural Network Classification with Uncertainty Quantification. Abstract: We propose a Gaussian confidence measure and its optimization, for use in neural network classifiers. The measure comes with theoretical results, simultaneously resolving two pressing problems in NN classification: uncertainty quantification, and robustness. Existing research in uncertainty quantification mostly revolves around the confidence reflected in the input feature space. Instead, we focus on the learned representation of the network and analyze the confidence in the penultimate layer space. We formally prove that, independent of optimization-procedural effects, a set of centroids always exists such that softmax classifiers are nearest-centroid classifiers. Softmax confidence, however, does not reflect that the classification is based on nearest centroids: artificially inflated confidence is also given to out-of-distributions samples that are not near any centroid, but slightly less distant from one centroid than from the others. Our new confidence measure is centroid-based, and hence no longer suffers from the artificial confidence inflation of out-of-distribution samples. We also show that our proposed centroidal confidence measure is providing a robustness certificate against attacks. As such, it manages to reflect what the model doesn't know (as demanded by uncertainty quantification), and to resolve the issue of robustness of neural networks.