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Traditional search methods try to obtain the most relevant information and rank it according to the degree of similarity to the queries. Diversity in query results is also preferred by a variety of applications since results very similar to each other cannot capture all aspects of the queried topic. In this paper, we focus on the lambda-diverse k-nearest neighbor search problem on spatial and multidimensional data. Unlike the approach of diversifying query results in a postprocessing step, we naturally obtain diverse results with the proposed geometric and index-based methods. We first make an analogy with the concept of Natural Neighbors (NatN) and propose a natural neighbor-based method for 2D and 3D data and an incremental browsing algorithm based on Gabriel graphs for higher dimensional spaces. We then introduce a diverse browsing method based on the distance browsing feature of spatial index structures, such as R-trees. The algorithm maintains a Priority Queue with mindivdist of the objects depending on both relevancy and angular diversity and efficiently prunes nondiverse items and nodes. We experiment with a number of spatial and high-dimensional data sets, including Factual's (http://www.factual.com/) US points-of-interest data set of 13M entries. On the experimental setup, the diverse browsing method is shown to be more efficient (regarding disk accesses) than k-NN search on R-trees, and more effective (regarding Maximal Marginal Relevance (MMR)) than the diverse nearest neighbor search techniques found in the literature.