I am kind of TAing for a class of real analysis, and I would like to speak a little about convex sets tomorrow, and explain why they are important. What kind of examples could I give? I was thinking of speaking a little about convex optimization, but as it is really out of my field of knowledge I can only give a very superficial overview of the subject.
Another possibility would be speaking of the Hahn-Banach separation theorem and of Krein-Milman's theorem, but I'm afraid it would be too complicated for them, and maybe not so interesting.
Do anyone have any good alternative ideas?