Convexity is the study of convex objects in some real vector space. Convexity is an amazing topic: just from the fact that a set is convex you can deduce all manner of smoothness properties, intersection properties, inequalities, geometric conclusions, and more!

I have collected some convexity-related nodes below (yes, this is a metanode). Please /msg me with additions! (Better yet, add new convexity-related nodes and /msg me about them...). Please note that although convexity in infinite-dimensional spaces (Banach spaces) is also extremely important, thus far we do not appear to have anything on that topic. So almost everything below is finite dimensional.

Convexity metanode

Intersection properties
Infinite dimensional spaces