Hypervolumes are used widely to conceptualize niches and trait distributions for both species and communities. Some hypervolumes are expected to be convex, with boundaries defined by only upper and lower limits (e.g., fundamental niches), while others are expected to be maximal, with boundaries defined by the limits of available space (e.g., potential niches). However, observed hypervolumes (e.g., realized niches) could also have holes, defined as unoccupied hyperspace representing deviations from these expectations that may indicate unconsidered ecological or evolutionary processes. Detecting holes in more than two dimensions has to date not been possible. I develop a mathematical approach, implemented in the hypervolume R package, to infer holes in large and high-dimensional data sets. As a demonstration analysis, I assess evidence for vacant niches in a Galapagos finch community on Isabela Island. These mathematical concepts and software tools for detecting holes provide approaches for addressing contemporary research questions across ecology and evolutionary biology.