Statistics - Machine Learning, Computer Science - Discrete Mathematics, Computer Science - Machine Learning, Computer Science - Symbolic Computation, Mathematics - Combinatorics, Mathematics - Functional Analysis

In this paper we investigate which subsets of the real plane are realisable as the set of points on which a one-layer ReLU neural network takes a positive value. In the case of cones we give a full characterisation of such sets. Furthermore, we give a necessary condition for any subset of $\mathbb R^d$. We give various examples of such one-layer neural networks.