Barany’s theorem states that such a colorful simplex must exist. Or, more formally: If there are d+1 colors each containing 0 in its convex hull, then a colorful simplex exists with 0 in its convex hull. This can be proved by first describing an algorithm that will find such a colorful simplex. We start by picking any colorful simplex T. In our 2-dimensional example zero is obviously not in the convex hull. If it were then we would be done.