TITLE: Why is the empty set convex?
QUESTION [5 upvotes]: Why is it the empty set, trivially convex? I see this results stated into a proof as something known, but I do not understand what's the idea idea behind it. How could I reason about convex combinations if the set has no elements?

REPLY [9 votes]: Here's another approach, without thinking about the formal quantifier logic of it.
The intersection of any two convex sets is convex, yes? Well, what's the intersection of two disjoint circles?