Computating the width of a set

Given a set of points <italic>P</italic> = {<italic>p</italic><subscrpt>1</subscrpt>,<italic>p</italic><subscrpt>2</subscrpt>,…,<italic>p<subscrpt>n</subscrpt></italic>} in three dimensions, the width of <italic>P, W</italic> (<italic>P</italic>), is defined as the minimum distance between parallel planes of support of <italic>P</italic>. It is shown that <italic>W</italic>(<italic>P</italic>) can be computed in <italic>&Ogr;</italic>(<italic>n</italic> log <italic>n</italic> + <italic>I</italic>) time and <italic>&Ogr;</italic>(<italic>n</italic>) space, where <italic>I</italic> is the number of antipodal pairs of edges of the convex hull of <italic>P</italic>, and in the worst case <italic>I</italic> - <italic>&Ogr;</italic>(<italic>n</italic><supscrpt>2</supscrpt>). If <italic>P</italic> is a set of points in the plane, this complexity can be reduced to <italic>&Ogr;</italic>(<italic>n</italic> log <italic>n</italic>). Finally, for simple polygons linear time suffices.