In automatic summarization, centrality-as-relevance means that the most important content of an information source, or a collection of information sources, corresponds to the most central passages, considering a representation where such notion makes sense (graph, spatial, etc.). We assess the main paradigms, and introduce a new centrality-based relevance model for automatic summarization that relies on the use of support sets to better estimate the relevant content. Geometric proximity is used to compute semantic relatedness. Centrality (relevance) is determined by considering the whole input source (and not only local information), and by taking into account the existence of minor topics or lateral subjects in the information sources to be summarized. The method consists in creating, for each passage of the input source, a support set consisting only of the most semantically related passages. Then, the determination of the most relevant content is achieved by selecting the passages that occur in the largest number of support sets. This model produces extractive summaries that are generic, and language- and domain-independent. Thorough automatic evaluation shows that the method achieves state-of-the-art performance, both in written text, and automatically transcribed speech summarization, including when compared to considerably more complex approaches.