Let $M$ be a compact codimension one submanifold of $\mathbb{R}^{n}$ which does not contaion $0$ and the origin lies in the bounded component of$\mathbb{R}^{n}-\{0\}$.
Is it true to say that:
For every $k>1$, there are (distinct) points $z_{1},z_{2},\ldots,z_{k}$ in M with $\sum_{i=1}^k z_{i}=0$
There are two motivations for this question:
1.For a closed curve $\gamma$ in the plane which surrounds origin, the geometric intuition says that this statement is likely true :a polygon with vertex on $\gamma$ with centroid at $0$.
- When the manifold $M$ is a retract of $\mathbb{R}^{n}-\{0\}$ with $\pi_{j}(M)\simeq \mathbb{Z}$ for some $j$, the statement is true for all $k$.
The reason is the following: Let $g:\mathbb{R}^{n}-\{0\}\to M$ be a retraction, then $g(z_{1}+z_{2}+\ldots +z_{k}/k)$$g((z_{1}+z_{2}+\ldots +z_{k})/k)$ is a Mean(see the reference below) on $M$ which contradic to page $391$ of the following paper by B. Eckmann "Social Choice and Topology A Case of Pure and Applied Mathematics, Expo. Math 2004, 385-393
http://www.sciencedirect.com/science/article/pii/S0723086904800161