Convex open subset as the domain required for convex optimisation

Question

Can we help unpack this statement from my Mathematical Economics course pack:

Let $f$ be a continuous twice differentiable function whose domain is a convex open subset $U$ of $\mathbb{R}^n$. If $f$ is a concave function on U and $Df(x_0) = 0$ then $x_0$ is a global maximum of $f$ on $U$

The first sentence is the part I'm struggling with, primarily the "Convex open subset" and I want to understand both the intuition and the technical mechanics behind the intuition. My understanding so far:

1. Continuous: This is necessary for our function to be differentiable across the entire domain so that we can find our stationary point.

2. Twice Differentiable: I'm assuming this is so we can access the second derivative for a hessian test to check if the function is indeed concave or not.

3. Domain is Convex: I understand this is a condition for convex optimisation, and I can kind of appreciate intuitively that if the domain wasn't a convex set. e.g. it was a donut. Then the function could potentially be discontinuous. But I feel like there is more to it, and maybe my explanation is wrong/not even close.

4. Domain is an Open Subset: This I really don't understand. How is an open subset going to help us vs a closed subset? All I can think of is something to do with maximum values at the edge of the domain, but I would have thought a closed subset would make more sense for this. (In the same way that the set of feasible vectors for our Lagrangian needs to be compact i.e. closed).

• Also given that the feasible set for a constrained optimisation problem is within or on the boundary of the constraint, and the constraint is presumably a subset of the domain, I'm even less clear about this condition on the domain of $f$.

I have searched Quora, Stack, YouTube and my Textbooks...And am struggling to come up with satisfactory answers. Apologies if this is basic or I have missed a post, but I will be immensely grateful for some thoughts!

Thanks!

Answer 1

The open part should be fairly intuitive if you unpack the definition of the $Df$ map; you need to have a 'neighbourhood' around a point if you want to know how the function changes in all directions near that. Convexity is there most likely to ensure that to find how it changes near a given point, you can study it restricted on lines joining that point and any other point in the set. Note that convexity is a nice sufficient condition that ensure that your manipulations are meaningful but most likely they are not necessary/ the most general conditions