I am working on a microeconomics problem, but I have just kind of just boiled down to the following problem involving convex sets. I have a convex set of vectors in $\mathbb{R^n_+}$ of the form $\{x:u(x)\geq t \}$ where $u(x)$ is some function whose level curves are convex to the origin, and this set intersects with the set of vectors $\{x\in \mathbb{R^n_+}:p\cdot x\leq w \}$ for a given vector $p\in \mathbb{R^n_+}$ and some positive real number $w$. Let's call the intersection $B$. I need to show there exists exactly one vector in $B$ that maximizes the value of $x_1$. In $\mathbb{R^2_+}$ you can see the problem easily, and it is easily solved, but I am having big problems trying to generalize the solution to anything higher than $\mathbb{R^2_+}$. I've drawn a lovely picture for you of the problem in $\mathbb{R^2_+}$:
Like I said, I figured out a proof pretty easily in $\mathbb{R^2_+}$, but it falls apart in higher dimensions. I've figured out a few things that I think might be relevant. I know that $B$ is itself convex. I feel like I should be able to use this fact to prove it but I am not 100% sure how. I know that the problem is equivalent to finding the vector in $B$ which maximizes the length of the projection onto the $x_1$ axis. I'm not sure if this is a profitable way to think about it.
Does anyone have any advice, resources, anything that I can look at to try to prove this?
Thanks immensely, and let me know if I've left out any important information.