Polygons with centroid at origin and vertices on compact codimension one submanifold of $\mathbb{R}^{n}-\{0\}$

Question

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 $0$.

2. 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 $g((z_{1}+z_{2}+\ldots +z_{k})/k)$ 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

Answer 1

Maybe I am missing something, but aren't you answering your own question? By the Eckmann theorem you cite, the result is true for a plane curve (since the hypotheses obviously hold). By @JeffStrom's comment, once you intersect with a plane, the intersection will contain a plane curve surrounding the origin.