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 for all points $$x_i$$ in the set,

is also in the set as long as

(2) is necessary to ensure that the summation in (1) is just a weighted average of the values of the points, otherwise we could always define sets that were outside the initial set just by multiplying the points under consideration by arbitrarily large values. It’s easy to see that while the set of points defining a sphere is convex, the set of points defining a torus—a donut shape—is not. Two points can be inside the set, while their weighted average--the line connecting them--is outside the set (think of two points on either side of the “hole” in the middle of a donut, for instance).

Why is this particular sort of geometric structure relevant to our discussion here? What is it about sets that behave like spheres rather than like donuts that make them more well-behaved mathematical representations of physical systems? We’ll return to that question in just a moment, but first let’s briefly examine the other way of articulating the linearity condition—(2) described above. Ultimately, we shall see that these two conditions are, at least in most cases of relevance to us, just different ways of looking at the same phenomenon. For the moment, though, it is dialectically useful to examine each of the two approaches on its own. 156