Assume that $n>1$.

The  configuration space of  $S^n$ is  defined as follows $$M_n=\{(x,y)\in S^n\times S^n\mid x \neq y\}$$

We  have  two  questions:

>1.Is there a  continuous  function $f:M_n \to S^{n-1}$  with $f(y,x)=-f(x,y)$, for all $x,y \in $S^{n}$?

>2.Is there  a  continuos  function $h: M_n \to \mathbb{R}^n$  such that $h(x,y)=-h(y,x)  $  and $h(x,-x)\neq 0$  for  all  $x,y \in S^n$

If the  answer to  either of these two questions is  "affirmative ", then we can provide an alternative  proof  for  the  Borsuk Ulam theorem, inductively. Because  an  equivalent  formulation of the  Borsuk Ulam theorem is that:

>There is  no  an odd continuous  function $g:S^{n+1}\to S^n$


Assuming that the  answer to either of  the  above two questions is  affirmative, we give a proof  for  the  equivalent  formulation of the  Borsuk Ulam theorem as follows:

Assume  that $g:S^{n+1}\to S^n$ is  an odd continuous  function. then $f(g(x),g(-x))$ ( or  $h(g(x),g(-x))$ ) is  an odd  continuous  function from $S^{n+1}$ to $S^{n-1}$ ( or to $\mathbb{R}^n \setminus \{0\}$). This  obviously  gives  a  contradiction by induction.