TITLE: Is there a simple proof of Borsuk-Ulam, given Brouwer?
QUESTION [9 upvotes]: (Brouwer) Any continuous function from a convex compact subset K of a Euclidian space to itself has a fixed point.

Given this lemma, is there a simple proof of:

(Borsuk-Ulam) Any continuous function $f \, : \, S^n \to R^n$ (where $S^n$ is the $n$-sphere) has a point $x$ for which $f(x) = f(-x)$.

?

REPLY [2 votes]: It appears that Borsuk-Ulam is strictly harder than Brouwer. Quote from Using the Borsuk-Ulam Theorem : Lectures on Topological Methods in Combinatorics and Geometry:
"It is instructive to compare this with the Brouwer fixed point theorem (...). The statement of the Borsuk–Ulam theorem sounds similar (and actually, it easily implies the Brouwer theorem; see below). But it involves an extra ingredient besides the topology of the considered spaces: a certain symmetry of these spaces, namely, the symmetry given by
the mapping $x \mapsto −x$ (which is often called the antipodality on $S^n$ and on $\Bbb R^n$)."
The exact strenght of Brouwer is known: is equivalent to WKL$_0$ over RCA$_0$ (Simpson, Subsystems of Second Order Arithmetic) but Borsuk-Ulam does not appear in the index of Subsystems...
UPDATE: a "elementary" proof of Borsuk-Ulam.