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.

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.

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 $i,j$ (my particular function does not seem to be well-defined otherwise). 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?)

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.