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The following problem seems a very hard one, is it known? It has a resemblance to the lonely runner conjecture. I am guessing.

In the plane let $v_i$ be $n$ unit vectors no two of them are colinear. Take $P_0$ any point in the plane and construct a successive set of points $P_i$ such that for every $i$, $1\le i\le n$ the segments $P_{i-1}P_{i}=v_j$ for some $j$. Every vector $v_j$ is used once. Can we choose the vectors so that the obtained path from $P_0\to P_n$ is either a simple cycle or has no crossings (planar)?

Ps The conjecture is not mine (Dominique Chesneau's question)

The following problem seems a very hard one, is it known? It has a resemblance to the lonely runner conjecture. I am guessing.

In the plane let $v_i$ be $n$ unit vectors no two of them are colinear. Take $P_0$ any point in the plane and construct a successive set of points $P_i$ such that for every $i$, $1\le i\le n$ the segments $P_{i-1}P_{i}=v_j$ for some $j$. Every vector $v_j$ is used once. Can we choose the vectors so that the obtained path from $P_0\to P_n$ is either a simple cycle or has no crossings (planar)?

The following problem seems a very hard one, is it known? It has a resemblance to the lonely runner conjecture. I am guessing.

In the plane let $v_i$ be $n$ unit vectors no two of them are colinear. Take $P_0$ any point in the plane and construct a successive set of points $P_i$ such that for every $i$, $1\le i\le n$ the segments $P_{i-1}P_{i}=v_j$ for some $j$. Every vector $v_j$ is used once. Can we choose the vectors so that the obtained path from $P_0\to P_n$ is either a simple cycle or has no crossings (planar)?

The following problem seems a very hard one, is it known? It has a resemblance to the lonely runner conjecture. I am guessing.

In the plane let $v_i$ be $n$ unit vectors no two of them are colinear. Take $P_0$ any point in the plane and construct a successive set of points $P_i$ such that for every $i$, $1\le i\le n$ the segments $P_{i-1}P_{i}=v_j$ for some $j$. Every vector $v_j$ is used once. Can we choose the vectors so that the obtained path from $P_0\to P_n$ is either a simple cycle or has no crossings (planar)?

Ps The conjecture is not mine (Dominique Chesneau's question)

The following problem seems a very hard one, is it known? It has a resemblance to the lonely runner conjecture. I am guessing.

In the plane let $v_i$ be $n$ unit vectors no two of them are colinear. Take $P_0$ any point in the plane and construct a successive set of points $P_i$ such that for every $i$, $1\le i\le n$ the segments $P_{i-1}P_{i}=v_j$ for some $j$. Every vector $v_j$ is used once. Can we choose the vectors so that the obtained path from $P_0\to P_n$ has no crossings?

The following problem seems a very hard one, is it known? It has a resemblance to the lonely runner conjecture. I am guessing.

In the plane let $v_i$ be $n$ unit vectors no two of them are colinear. Take $P_0$ any point in the plane and construct a successive set of points $P_i$ such that for every $i$, $1\le i\le n$ the segments $P_{i-1}P_{i}=v_j$ for some $j$. Every vector $v_j$ is used once. Can we choose the vectors so that the obtained path from $P_0\to P_n$ is either a simple cycle or has no crossings (planar)?