TITLE: Borsuk pairs of Banach spaces 
QUESTION [9 upvotes]: Given $X$, $Y$ two real Banach spaces, let's say that $(X,\ Y)$
is a Borsuk pair if for any continuous mapping $T$ : {$x$ $\in$
$X$ ; $||x||\leq1$} $\rightarrow$ $Y$ s.t. $T$ is odd on {$x$
$\in$ $X$ ; $||x||=1$}, it follows that the set $T^{-1}$( { 0 } )
is nonempty.
My conjecture is : $(X,\ Y)$ Borsuk pair $\Longleftrightarrow$
dim $Y$ < $\infty$ and dim $Y$ $\leq$ dim $X$. [Here "dim"
stands for the algebraic dimension; it may be a natural number or $\infty$.]
In other words, there is no Borsuk pair with $X$, $Y$ both infinite-dimensional.
Any thoughts ?
[There exist several infinite-dimensional versions of the Classical Borsuk Theorem, e.g.,
when $X$ is reflexive, $Y$ is its dual, and $T$ is a demicontinuous mapping of monotone type.]
[I think the "big" problem (if it truly exists, of course ;-)) would be when both $X$, $Y$ are  infinite dimensional,
and the density character dens($X$) $\gg$ dens($Y$), i.e., when $X$ is "huge".]

REPLY [2 votes]: This is a  negative result, heavily relying on P. Dodos'
answer to Boundedness of nonlinear continuous functionals.
Let $\kappa$ be a measurable cardinal, let $K$ be the closed unit
ball of $\ell_{\infty}\left(\kappa\right)$, and suppose, ad absurdum,
that there exists a continuous mapping $F: K \rightarrow$ 
$\ell^{1}
$\ {${ 0} $}
that   is odd on the unit sphere of 
$\ell_{\infty}\left(\kappa\right)$.
Let $j:\{\ell}^{1}$ 
    $\rightarrow$ 
${\ell}^{2}$ 
be
the natural injection, 
and let $\rho$ be a metric generating the weak
topology on the closed unit ball of ${\ell}^{2}$. Define
next the continuous operator $T:K$ $\rightarrow$
 ${\ell}^{2}$\ {${ 0} $} by
$Tx:=$ $\dfrac{jFx}{\left\Vert jFx\right\Vert _{2}}$. 
Define also
the continuous function $f:\ K\rightarrow\mathbb{R}$,
expressed
by $f(x):=$ $\rho\left(0,Tx\right)^{-1}$. Then there is a subspace
$E$ of $\ell_{\infty}\left(\kappa\right)$ that is isomorphic
to $c_{0}\left(\kappa\right)$ s.t. $f(K$ $\cap$ $E$ $)$ is
bounded. Let $\left(P_{n}\right)$ be a sequence of ortho-projectors
on ${\ell}^{2}$, having $n$-dimensional ranges $Y_{n}$,
and s.t. $P_{n}$(x)$\ \rightarrow\ x $ 
for each $x$ in ${\ell}^{2}$.
Take also a sequence $\left(X_{n}\right)$ of $n$-dimensional subspaces
of $E$, and apply the classical Borsuk Theorem for the mappings 
$T_{n}$:=$P_{n}T:K\cap X_{n}$ $\rightarrow Y_{n}$,
in order to get a sequence $\left(x_{n}\right)$ s.t.   $T_{n}$ $x_{n}=$0$.$
Then it is easy to show that 
$Tx_{n}$ $\rightharpoonup0$ [i.e.,
weakly] in ${\ell}^{2}$, hence $f$ is not bounded on $K\cap E$,
a contradiction.
Therefore, the Conjecture is not true.