In convex optimization, we usually try to solve a problem of minimizing/maximizing a convex function over a convex set/domain. Intuitively, I think it makes sense but I'm not be able to point out the exact reason why we want the set to be convex. Most of the analysis that I've seen so far usually use the convexity of the objective function rather than the set.
Why is the convexity of the domain important?
What happens if we optimize a convex function over non-convex set?
Any insights is appreciated!