Prove that for any convex polyhedron, there exist 4 vertices A, B, C, D with the following property: for each of the four vertices A, B, C, D, the polyhedron lies entirely on one side of the plane passing through that point and parallel to the plane passing through the other three vertices.
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ABCD - a tetrahedron of the largest volume among all possible tetrahedra formed by the vertices of the polyhedron.
## Solution
Among all quadruples of vertices of the polyhedron, we select the quadruple A, B, C, D, forming a tetrahedron of the largest volume. We will prove that the quadruple of vertices A, B, C, D satisfies the condition of the problem. Consider, for example, the plane P passing through vertex A and parallel to the plane passing through vertices B, C, and D. Suppose that not the entire polyhedron lies on one side of plane P. Then there exists some vertex M of the polyhedron located on different sides of plane P from the plane BCD. Consider the tetrahedron MBCD. It has the same base BCD as the tetrahedron ABCD, but the height dropped from vertex M to the base is greater than the height of the tetrahedron ABCD dropped from vertex A to the base. This implies that the volume of the tetrahedron MBCD is greater than the volume of the tetrahedron ABCD, contrary to the selection of vertices A, B, C, D. The contradiction obtained shows that the entire polyhedron lies on one side of plane P. Similar reasoning proves that the polyhedron lies entirely on one side of the other three planes mentioned in the condition.
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