In the 1954 paper Continuous Functions From Spheres to Euclidean Spaces, author Chung-Tao Yang cites the following problem:
Problem 1: Given a (continuous) map $f$ of an $(m+n-2)$-sphere $S^{m+n-2}$ into $\mathbb{R}^m$ and $n$ distinct points $y_1,\ldots,y_n\in S^{m+n-2}$, do there exist $n$ points of $S^{m+n-2}$ which are congruent to $y_1,\ldots, y_n$ and are mapped into a single point by $f$?
As of the writing of the paper, the problem was apparently open, with only scattered partial results shown:
The case $n=2,m=1$ is very easy to resolve using the intermediate value theorem.
The case $n=2$ where the points are antipodal is just the Borsuk–Ulam theorem; I expect it has been generalized to other distances between $y_1$ and $y_2$, but I don't know of a source offhand.
In 1950, Yamabe-Yujobô generalized a result of Kakutani to show the result in the case where $m=1$ and the $y_i$ are orthogonal.
In 1955, E. E. Floyd showed the result for $n=3,m=1$.
I believe that I can use the basic argument from the paper of Yamabe and Yujobô to show that in the $m=1$ case, the result for $y_1,\ldots,y_n$ on $S^{n-1}$ implies the result for $y_1,\ldots,y_{n+1}$ on $S^n$ if $y_{n+1}$ is of equal distance to all previous $y_i$. (This for instance gives the result for every isosceles triangle on $S^2$, as Floyd notes was apparently done in a master's thesis of R.D. Johnson.)
I'm curious about further progress that has been made on the problem in the last 60 years; lacking a nice term to search for this conjecture under, I haven't turned up much, but I'd be surprised if there weren't new results or indeed if it had been fully resolved. I'm particularly curious about the $m=1$ case, but would be interested to hear about any pieces of the problem.
