# Multi-podal points

Two points $x,y \in \mathbb{R}^n$ are called antipodal if $x = -y$.

Stated differently, $x,y$ are antipodal if:

• They have the same absolute value in each of their $n$ coordinates;
• Each of their non-zero coordinates appears with a positive sign exactly once (either in $x$ or in $y$).

The Borsuk-Ulam theorem is a well-known theorem about antipodal points.

Let's call a set of points $\{x_i\}_{i=1}^k \subseteq \mathbb{R}^n$ multipodal if:

• They have the same absolute value in each of their $n$ coordinates;
• Each of their non-zero coordinates appears with a positive sign exactly once.

Every pair of antipodal points is a multipodal set of size 2. Here are some multipodal sets of size 3:

• $(+1, -2, -3); (-1, +2, -3); (-1, -2, +3)$
• $(+1, -2, 0, -3, -2); (-1, +2, 0, -3, -2); (-1, -2, 0, +3, +2)$
• $(+1,+2); (-1,-2); (-1,-2)$
• $(-1,+2); (+1,-2); (-1,-2)$

Let's call a function $f$ k-podal if, for every multipodal set $X$ of size $k$:

$$\sum_{x\in X}f(x) = 0$$

In particular, a 2-podal function is just another name for an odd function: $f(-x)=-f(x)$.

I am looking for information about these "multipodal points" and "$k$-podal" functions. In particular:

• Is there another, standard term for them?
• What do we know about the degree of "$k$-podal" functions, for $k\geq 3$? E.g, is there a theorem saying that every $k$-podal function from $S^n$ to $S^n$ has an odd degree? (this is known for $k=2$).
• The following theorem is true for $k=2$ - it is the Borsuk-Ulam theorem:

Every continuous $k$-podal function $f: S^n \to \mathbb{R}^n$ has a zero.

Is it correct for any $k\geq 3$?

• if $a,b,c,...>0, (-a,-b,-c,...)$ could always be present. For a function, let $a$ be the positive component, then $f(a,b,c)\to$ either $f(-a,-b,c)$ or $f(-a,b,-c)$, but this won't suffice as a definition. – JonMark Perry May 25 '15 at 17:55
• @JonMarkPerry Excuse me but I didn't understand you... what do you mean by "could always be present"? What do you mean by "won't suffice as a definition"? – Erel Segal-Halevi May 26 '15 at 4:28
• a set of anti-podal points contains all possible members, by your definition, we can have say (1,-2), (-1,2), (-1,-2). the function bit i think i got wrong, but here's my proof of your Q, $|S^n|=|\mathbb{R}^n \cup \{\infty\}|$, so by the Pigeonhole principle, there is a point of $|S^n|$ unaccounted for in any mapping to $\mathbb{R}^n$. – JonMark Perry May 26 '15 at 4:40