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This problem can be solved in time O(nc) by combining a weight-balanced tree of the convex hull vertices with a farthest neighbor data structure of Agarwal and Matoušek .
2005, A.D. Alexandrov, Convex Polyhedra, page 23:
Thus, the polyhedron and the convex hull of the set of its vertices coincide, which was to be proved. Since the convex hull of a set M is determined uniquely from M, Theorem 1 implies that a bounded solid polyhedron and its surface, i.e., a closed convex polyhedron, are determined uniquely from its vertices.
2007, Charalambos D. Aliprantis, Kim C. Border, Infinite Dimensional Analysis: A Hitchhiker's Guide, page 185:
The convex hull of a compact subset of an infinite dimensional topological vector space need not be a compact set.
Convex geometry on Wikipedia.Wikipedia 