How is every affine set convex?

Very intuitively, an Affine set contains a line through any two distinct points in the set. This means that from a purely geometric point of view, affine sets are lines, planes, 3D planes ... which are clearly also Convex set. For a neat mathy explanation check Boyd's notes http://www.stanford.edu/~boyd/

I am guessing that you find it surprising because it appears as if the condition for convexity is stronger, and the reverse implication should have been true. So, perhaps you are asking how can something that satisfies simpler conditions (affine sets) actually turn out to satisfy something that involves a stronger condition (convexity)? If indeed this is your dilemma,  I will attempt to shine light on this. Stating your equations in words, an affine set requires that every affine combination of its points (weights adding to unity) lies in the set, whereas a convex set requires that every convex combination of its points (nonnegative weights adding to unity) lie in the set. Now, an affine combination is a simpler condition than convex combination and as a consequence the set of all convex combinations of a collection of points (which is a stronger condition) is a strict subset of all affine combinations of the same collection. So, if you require that every affine combination of points belong to the set, you are also requiring that every convex combination belongs to the set. Thus, affine sets are convex. P.S: The set of all affine combinations of two points is the line joining the two points and the set of all convex combinations of two points is the line segment joining the two points. This is the basis of the geometric intuition provided by all the other answers. I wanted to provide a complementary view by taking a wild guess at the perspective you may be looking for.

Rather than discussing lines and sets at all, let's examine the very basics of the problem at hand. The way that you've defined the problem actually helps tremendously. The key thing you're missing is that the definitions of affine and convex are for all w and for all w such that [math]w_i > 0[/math] for all i. That means that in order for a set to be affine and not convex, w must be chosen from the positive reals yet not belong to the reals. (admittedly not rigorous in notation) QED.