I'm working on a project about the two dimensional Borsuk-Ulam theorem (any given continuous function of a sphere into a two dimensional Euclidean space maps some pair of antipodal points on the same point). The general idea is to try to find an algorithm that finds those points (or an approximation of those points). My problem is that the proof of this theorem I've read (which is the "classic one" I think) is non-constructive (it doesn't "give" a natural algorithm to find those points).
However, I'm not sure there is an algorithm to solve my problem "in general", because I've read (I'm not sure of the source) that there is no better solution to the "stolen necklace" problem than the "naive" one, and the existence of solution to this problem is a consequence of the BU Theorem (finding the "antipodal points" is equivalent to solving the problem). But I thought that maybe in some particular cases (if the function is differentiable for instance) there could be some algorithms that could help.
To be more precise, I thought that since finding a point $x$ such that $f(x)=f(-x)$ is the same as finding $g(x)=0$ where $g(x)=f(x)-f(-x)$, there was probably some method to find the zeros of a function $\mathbb{R}^3\to\mathbb{R}^2$ under some conditions.
Does anyone know about such a method? And can you think of any other particular case in which an efficient algorithm to find those points could exist?