A general approach to d-dimensional geometric queries

It is shown that any bounded region in <italic>E</italic><supscrpt><italic>d</italic></supscrpt> can be divided into 2<supscrpt><italic>d</italic></supscrpt> subregions of equal volume in such a way that no hyperplane in <italic>E</italic><supscrpt><italic>d</italic></supscrpt> can intersect all 2<supscrpt><italic>d</italic></supscrpt> of the subregions. This theorem provides the basis of a data structure scheme for organizing <italic>n</italic> points in <italic>d</italic> dimensions. Under this scheme, a broad class of geometric queries in <italic>d</italic> dimensions, including many common problems in range search and optimization, can be solved in linear storage space and sublinear time.