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Let $C\subset \mathbb{R}^2$ be a smooth, simple closed curve. The thief is inside $C$. Before he starts to move, the police bureau of the $\mathbb{R}^2$ world can freely place countably infinite officers on $C$. We know that

• The thief and the officers move simultaneously and continuously. Maximum speed is $1$ for everyone.
• The officers are restricted to move on $C$. They can pass right through each other without collision.
• The thief is caught if his coordinates coincide with those of an officer.

There're 3 possibilities:

1. The thief always has a plan to get out of $C$.
2. The officers always have a plan to prevent the thief from getting out.
3. It depends on the shape of $C$.

Which one is true?

Response to comments:

1. If $m(t)$ is a path of the thief, continuous movement means $\vert m(t)-m(s)\vert \leq \vert t-s\vert, \forall t,s$. In particular, we do not require the path to be differentiable. Similarly for an officer path $l(t)$.
2. Let $Q(t)\subset C$ be the set of officers at time $t$. The thief escapes if $\exists t$ such that $m(t)\cap C/Q(t)\neq \emptyset$.
3. The thief and the officers have perfect information about each other's current positions, and move according to that information. For example, officer 1 may adopt a strategy like "if $\vert m_x-l_x\vert\gt 0$, move at maximum speed in the direction that decreases it, otherwise stay still".

Let $C\subset \mathbb{R}^2$ be a smooth, simple closed curve. The thief is inside $C$. Before he starts to move, the police bureau of the $\mathbb{R}^2$ world can freely place countably infinite officers on $C$. We know that

• The thief and the officers move simultaneously and continuously. Maximum speed is $1$ for everyone.
• The officers are restricted to move on $C$. They can pass right through each other without collision.
• The thief is caught if his coordinates coincide with those of an officer.

There're 3 possibilities:

1. The thief always has a plan to get out of $C$.
2. The officers always have a plan to prevent the thief from getting out.
3. It depends on the shape of $C$.

Which one is true?

Response to comments:

1. If $m(t)$ is a path of the thief, continuous movement means $\vert m(t)-m(s)\vert \leq \vert t-s\vert, \forall t,s$. In particular, we do not require the path to be differentiable. Similarly for an officer path $l(t)$.
2. Let $Q(t)\subset C$ be the set of officers at time $t$. The thief escapes if $\ \exists_t\,m(t)\,\in\, C\setminus Q(t)$.
3. The thief and the officers have perfect information about everybody's current position and move according to that information. For example, officer 1 may adopt a strategy like "if $\vert m_x-l_x\vert\gt 0$, move at maximum speed in the direction that decreases it, otherwise stay still".

Let $C\subset \mathbb{R}^2$ be a smooth, simple closed curve. The thief is inside $C$. Before he starts to move, the police bureau of the $\mathbb{R}^2$ world can freely place countably infinite officers on $C$. We know that

• The thief and the officers move simultaneously and continuously. Maximum speed is $1$ for everyone.
• The officers are restricted to move on $C$. They can pass right through each other without collision.
• The thief is caught if his coordinates coincide with those of an officer.

There're 3 possibilities:

1. The thief always has a plan to get out of $C$.
2. The officers always have a plan to prevent the thief from getting out.
3. It depends on the shape of $C$.

Which one is true?

Response to comments:

1. If $m(t)$ is a path of the thief, continuous movement means $\vert m(t)-m(s)\vert \leq \vert t-s\vert, \forall t,s$. In particular, we do not require the path to be differentiable. Similarly for an officer path $l(t)$.
2. Let $Q(t)\subset C$ be the set of officers at time $t$. The thief escapes if $\exists t$ such that $m(t)\cap C/Q(t)\neq \emptyset$.
3. The thief and the officers have perfect information about each other's current positions, and move according to that information. For example, officer 1 may adopt a strategy like "if $m_x-l_x\gt 0$, move at maximum speed in the direction that decreases it, otherwise stay still".

Let $C\subset \mathbb{R}^2$ be a smooth, simple closed curve. The thief is inside $C$. Before he starts to move, the police bureau of the $\mathbb{R}^2$ world can freely place countably infinite officers on $C$. We know that

• The thief and the officers move simultaneously and continuously. Maximum speed is $1$ for everyone.
• The officers are restricted to move on $C$. They can pass right through each other without collision.
• The thief is caught if his coordinates coincide with those of an officer.

There're 3 possibilities:

1. The thief always has a plan to get out of $C$.
2. The officers always have a plan to prevent the thief from getting out.
3. It depends on the shape of $C$.

Which one is true?

Response to comments:

1. If $m(t)$ is a path of the thief, continuous movement means $\vert m(t)-m(s)\vert \leq \vert t-s\vert, \forall t,s$. In particular, we do not require the path to be differentiable. Similarly for an officer path $l(t)$.
2. Let $Q(t)\subset C$ be the set of officers at time $t$. The thief escapes if $\exists t$ such that $m(t)\cap C/Q(t)\neq \emptyset$.
3. The thief and the officers have perfect information about each other's current positions, and move according to that information. For example, officer 1 may adopt a strategy like "if $\vert m_x-l_x\vert\gt 0$, move at maximum speed in the direction that decreases it, otherwise stay still".

Let $C\subset \mathbb{R}^2$ be a smooth, simple closed curve. The thief is inside $C$. Before he starts to move, the police bureau of the $\mathbb{R}^2$ world can freely place countably infinite officers on $C$. We know that

• The thief and the officers move simultaneously and continuously. Maximum speed is $1$ for everyone.
• The officers are restricted to move on $C$. They can pass right through each other without collision.
• The thief is caught if his coordinates coincide with those of an officer.

There're 3 possibilities:

1. The thief always has a plan to get out of $C$.
2. The officers always have a plan to prevent the thief from getting out.
3. It depends on the shape of $C$.

Which one is true?

Let $C\subset \mathbb{R}^2$ be a smooth, simple closed curve. The thief is inside $C$. Before he starts to move, the police bureau of the $\mathbb{R}^2$ world can freely place countably infinite officers on $C$. We know that

• The thief and the officers move simultaneously and continuously. Maximum speed is $1$ for everyone.
• The officers are restricted to move on $C$. They can pass right through each other without collision.
• The thief is caught if his coordinates coincide with those of an officer.

There're 3 possibilities:

1. The thief always has a plan to get out of $C$.
2. The officers always have a plan to prevent the thief from getting out.
3. It depends on the shape of $C$.

Which one is true?

Response to comments:

1. If $m(t)$ is a path of the thief, continuous movement means $\vert m(t)-m(s)\vert \leq \vert t-s\vert, \forall t,s$. In particular, we do not require the path to be differentiable. Similarly for an officer path $l(t)$.
2. Let $Q(t)\subset C$ be the set of officers at time $t$. The thief escapes if $\exists t$ such that $m(t)\cap C/Q(t)\neq \emptyset$.
3. The thief and the officers have perfect information about each other's current positions, and move according to that information. For example, officer 1 may adopt a strategy like "if $m_x-l_x\gt 0$, move at maximum speed in the direction that decreases it, otherwise stay still".