Linear programming and convex hulls made easy

We present two randomized algorithms. One solves linear programs involving <italic>m</italic> constraints in <italic>d</italic> variables in expected time <italic>&Ogr;</italic>(<italic>m</italic>). The other constructs convex hulls of <italic>n</italic> points in R<italic><supscrpt>d</supscrpt>, d</italic> > 3, in expected time <italic>&Ogr;</italic>(<italic>n</italic><supscrpt>⌈<italic>d</italic>/2⌉</supscrpt>). In both bounds <italic>d</italic> is considered to be a constant. In the linear programming algorithm the dependence of the time bound on <italic>d</italic> is of the form <italic>d</italic>!. The main virtue of our results lies in the utter simplicity of the algorithms as well as their analyses.