TITLE: Proving that a function's image contains (1/n,...,1/n)
QUESTION [26 upvotes]: This question is a follow-up to a previous question answered by Neil Strickland:
Map from simplex to itself that preserves sub-simplices
Let $B$ denote the closed unit ball in $\mathbb{R}^2$ and let $\Delta_{n-1}$ denote the $(n-1)$-simplex.  I have a continuous function $f(x_1,\dots,x_n):B^n \rightarrow \Delta_{n-1}$ defined for all subsets $\lbrace x_1,\dots,x_n\rbrace \subset B$ of size $n$ that satisfy $x_i \neq x_j$ for all pairs $i,j$ (in other words, the function is only defined if all of the $n$ arguments are distinct).  This function has the property that, if $\sigma$ denotes a permutation, then $f(\sigma(x_1,\dots,x_n)) = \sigma(f(x_1,\dots,x_n))$. In other words, permuting the arguments of the function merely permutes the output.  My question is:  are there non-trivial sufficient conditions on $f$ under which the point $(1/n , \dots, 1/n)$ lies in the image of this map?  (or, even better, is this always the case?)
Here's one property of the map $f$ that I can add regarding the requirement that arguments be distinct:  if $\lbrace \mathbf{x}_k \rbrace$ is a sequence of $n$-tuples (with distinct entries) in $B$ that converges to an $n$-tuple $\bar{\mathbf{x}}$ with (possibly) non-distinct entries, then the limit of $f(\mathbf{x}_k)$ exists if and only if, for each pair of entries $x_i^k$ and $x_j^k$ in the $n$-tuple, the unit direction vector from $x_i^k$ to $x_j^k$ (i.e. $\frac{x_i^k - x_j^k}{||x_i^k - x_j^k||}$) has a limit.

REPLY [12 votes]: Let me rephrase the question and give a complete answer:
What are conditions for the existence of a $\Sigma_n$-equivariant map
$$f: F_n(B)\to \Delta_{n-1}\backslash(\tfrac{1}{n},\dots,\tfrac{1}{n})\; ?$$
Here $B\subset \mathbb{R}^2$ denotes the closed unit ball, $F_n(B)$ is the ordered configuration space of $n$ points on $B$ with $\Sigma_n$ acting by permutation and $\Sigma_n$ acts on $\Delta_{n-1}:=\{(x_1,\dots,x_n)\in\mathbb{R}^n: \sum_ix_i=1, 0\leq x_i\leq 1\}$ by permuting the coordinates.
As Neil Strickland pointed out, using an equivariant homeomorphism, we can replace 
$\Delta_{n-1}\backslash(\tfrac{1}{n},\dots,\tfrac{1}{n})$ by $\partial\Delta_{n-1}\cong_{\Sigma_n} S^{n-2}.$ This sphere $S^{n-2}$ can be viewed as $S(W_n)$, where $W_n:=\{(x_1,\dots,x_n)\in\mathbb{R}^n: \sum_ix_i=0\}$ and $\Sigma_n$ is again acting by permuting the coordinates. Using equivariant homotopy equivalences, we can replace $F_n(B)$ by $F(\mathbb{R}^2)$ and further use an equivariant model of dimension $n-1$, which Neil Strickland calls $X_0$ and I will call $\mathcal{F}_n(\mathbb{R}^2)$. So an equivalent question is:
What are conditions for the existence of a $\Sigma_n$-equivariant map
$$f: \mathcal{F}_n(\mathbb{R}^2)\to S(W_n)\; ?$$
Fortunately, there is a complete answer to that questions; it is given by Blagojević and Ziegler: arxiv, springer:
Such a map exists if and only if $n\geq 2$ is a not a prime power
The authors use equivariant obstruction theory and an explicit model $\mathcal{F_n(\mathbb{R}^2)}$, which is called $\mathcal{F}(d,n)$ in their notation. Just plug in $d=2$ and the result is the main theorem of section 4 in the paper on the arxiv. 
From the nonexistence of the map we conclude that the barycenter $(\frac{1}{n},\dots,\frac{1}{n})$ is hit: If $n$ is a prime power, then your continous equivariant function $B^n\to\Delta_{n-1}$ will always hit the barycenter  and if $n$ is not a prime power, there exist equivariant functions that miss the barycenter.