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The cops should always win. Here is a sketch of a proof.

1. One cop (for short) trying to catch the thief crossing a segment: the cop can run on the segment at top speed towards the thief (provided the thief starts far enough and the cop's reaction time is 0). Clearly they will meet.

2. Likewise a dense set of cops on a convex curve $C$: they can run towards the thief only when the latter is "close" to them, while keeping dense all the time. To make this precise: let $r$ be the minimum radius of curvature, which exists because $C$ is smooth and compact; whenever the thief is closer to $C$ than $r$ there is a unique point $P\in C$ to which the thief is closest, with distance $d_P<r$; if $d\ge d_P$ is the distance of a cop from the thief, that cop should run towards $P$ at speed $\text{max}(1-d/r,0)$. This strategy is continuous on $C$ and guarantees that the cops will stay a dense set, and also approaches the strategy in 1) as the thief gets closer to $C$, guaranteeing an eventual catch.

3. This works on a non convex curve too. On a curve with a singularity the thief can try to split the cops and trap them in a cul de sac, but on a smooth curve that is thwarted by the cops remaining dense always.

The cops should always win. Here is a sketch of a proof.

1. One cop (for short) trying to catch the thief crossing a segment: the cop can run on the segment at top speed towards the thief (provided the thief starts far enough and the cop's reaction time is 0). Clearly they will meet.

2. Likewise a dense set of cops on a convex curve $C$: they can run towards the thief only when the latter is "close" to them, while keeping dense all the time. To make this precise: let $r$ be the minimum radius of curvature, which exists because $C$ is smooth and compact; whenever the thief is closer to $C$ than $r$ there is a unique point $P\in C$ to which the thief is closest, with distance $d_P<r$; if $d\ge d_P$ is the distance of a cop from the thief, that cop should run towards $P$ at speed $\text{max}(1-d/r,0)$. This strategy is continuous on $C$ and guarantees that the cops will stay a dense set, and also approaches the strategy in 1) as the thief gets closer to $C$, guaranteeing an eventual catch.

3. This works on a non convex curve too. On a curve with a singularity the thief can try to split the cops and trap them in a cul de sac, but on a smooth curve that is thwarted by the cops remaining dense always.

Update. See comments for where this argument goes wrong.

The cops should always win. Here is a (not completely rigorous) justification.

1. One cop (for short) trying to catch the thief crossing a segment: if the cop moves on the segment at top speed towards the thief (provided the thief starts far enough and the cop's reaction time is 0) clearly they will meet.

2. Likewise a dense set of cops on a convex curve $C$: they can run towards the thief only when the latter is closer to them than the minimum radius of curvature, while remaining dense all the time. To make this more precise, let $r$ be the minimum radius of curvature, which exists because $C$ is smooth and compact. Whenever the thief is closer to $C$ than $r$ there is a unique point $P\in C$ to which the thief is closest, with distance $d_P<r$. If $d\ge d_P$ is the distance of a cop from the thief, that cop should run towards $P$ at speed $\text{max}(1-d/r,0)$. This strategy is continuous on $C$ and guarantees that the cops will remain a dense set, and also approaches the strategy in 1) as the thief gets closer to $C$, guaranteeing an eventual catch.

3. This works on a non convex curve too. The thief may try to trap the cops in a cul de sac, but that is thwarted by the cops remaining dense always.

The cops should always win. Here is a sketch of a proof.

1. One cop (for short) trying to catch the thief crossing a segment: the cop can run on the segment at top speed towards the thief (provided the thief starts far enough and the cop's reaction time is 0). Clearly they will meet.

2. Likewise a dense set of cops on a convex curve $C$: they can run towards the thief only when the latter is "close" to them, while keeping dense all the time. To make this precise: let $r$ be the minimum radius of curvature, which exists because $C$ is smooth and compact; whenever the thief is closer to $C$ than $r$ there is a unique point $P\in C$ to which the thief is closest, with distance $d_P<r$; if $d\ge d_P$ is the distance of a cop from the thief, that cop should run towards $P$ at speed $\text{max}(1-d/r,0)$. This strategy is continuous on $C$ and guarantees that the cops will stay a dense set, and also approaches the strategy in 1) as the thief gets closer to $C$, guaranteeing an eventual catch.

3. This works on a non convex curve too. On a curve with a singularity the thief can try to split the cops and trap them in a cul de sac, but on a smooth curve that is thwarted by the cops remaining dense always.

The cops should always win. Here is a (not completely rigorous) justification.

1. One cop (for short) trying to catch the thief crossing a segment: if the cop moves on the segment at top speed towards the thief (provided the thief starts far enough and the cop's reaction time is 0) clearly they will meet.

2. Likewise a dense set of cops on a convex curve $C$: they can run towards the thief only when the latter is closer to them than the minimum radius of curvature, while remaining dense all the time. To make this more precise, let $r$ be the minimum radius of curvature, which exists because $C$ is smooth and compact. Whenever the thief is closer to $C$ than $r$ there is a unique point $P\in C$ to which the thief is closest, with distance $d_P<r$. If $d\ge d_P$ is the distance of a cop from the thief, that cop should run towards $P$ at speed $\text{min}(1,\text{max}(r-d,0))$. This strategy is continuous on $C$ and guarantees that the cops will remain a dense set, and also approaches the strategy in 1) as the thief gets closer to $C$, guaranteeing an eventual catch.

3. This works on a non convex curve too. The thief may try to trap the cops in a cul de sac, but that is thwarted by the cops remaining dense always.

The cops should always win. Here is a (not completely rigorous) justification.

1. One cop (for short) trying to catch the thief crossing a segment: if the cop moves on the segment at top speed towards the thief (provided the thief starts far enough and the cop's reaction time is 0) clearly they will meet.

2. Likewise a dense set of cops on a convex curve $C$: they can run towards the thief only when the latter is closer to them than the minimum radius of curvature, while remaining dense all the time. To make this more precise, let $r$ be the minimum radius of curvature, which exists because $C$ is smooth and compact. Whenever the thief is closer to $C$ than $r$ there is a unique point $P\in C$ to which the thief is closest, with distance $d_P<r$. If $d\ge d_P$ is the distance of a cop from the thief, that cop should run towards $P$ at speed $\text{max}(1-d/r,0)$. This strategy is continuous on $C$ and guarantees that the cops will remain a dense set, and also approaches the strategy in 1) as the thief gets closer to $C$, guaranteeing an eventual catch.

3. This works on a non convex curve too. The thief may try to trap the cops in a cul de sac, but that is thwarted by the cops remaining dense always.