Convexity (thing)

(Mathematics:)

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

Definition

• convex
• Helly's theorem; its proof
• star convex (not really convex, but close enough)
• interval (convexity in one dimension is boring)
• convex combination
• convex hull

Intersection properties

• Caratheodory's theorem; its proof
• Radon's theorem; its proof of Radon's theorem

Inequalities

• Jensen's inequality; its proof
• Brunn's inequality
• Brunn-Minkowski inequality (applies also to non-convex bodies, but convexity always gives the harder cases, so it fits in). Also contains the proof of Brunn's inequality.

Calculus

• differentials of a convex object exist almost everywhere; better yet,
• positive semidefinite, negative semidefinite, second derivative are all relevant when the object is smooth -- and a convex object is smooth almost everywhere...

Polyhedra

• Platonic solids are convex
• a convex polyhedron in R³ has a dual polyhedron

Infinite dimensional spaces

• locally convex space
• Banach spaces are locally convex
• So are Frechet spaces, in fact