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 $k\geq 3$?