Deterministic algorithms for 2-d convex programming and 3-d online linear programming

We present a deterministic algorithm for solving two-dimensional convex programs with a linear objective function. The algorithm requires O(k log k) primitive operations for k constraints; if a feasible point is given, the bound reduces to O(k log k/log log k). As a consequence, we can decide whether k convex n-gons in the plane have a common intersection in O(k log n min{log k, log log n}) worst-case time. Furthermore, we can solve the three-dimensional online linear programming problem in o(log(3) n) worst-case time per operation.