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Let me elaborate on Sam Hopkins' comment.

The main reason that makes this and other problems on continuous curves so hard is that a "simple closed curve" or "Jordan curve", i.e. a non-self-intersecting continuous loop in the plane, can be a horrible object, for instance a nowhere differentiable curve such as the Koch snowflake and other fractal curves. There are also Jordan curves of positive area (first constructed by Osgood in 1903).

In fact, as it is also explained in the Wikipedia article that you linked, the problem is actually solved for "well-behaved" curves, such as convex curves or piecewise analytic curves, i.e. for objects that are close to our intuitive notion of "continuous closed loop".

A possible strategy to solve the problem in the general case is to try to approximate your Jordan curve by using well behaved curves, for which we know that the result holds true, and then pass to the limit. The technical difficulty with this approach is that a limit of squares is not necessarily a square, but it can be a single point (i.e., a square "of side length 0"), so it is not clear whether it is always possible to take an approximating sequences of curves that works.

Let me elaborate on Sam Hopkins' comment.

The main reason that makes this and other problems on continuous curves so hard is that a "simple closed curve" or "Jordan curve", i.e. a non-self-intersecting continuous loop in the plane, can be a horrible object, for instance a nowhere differentiable curve such as the Koch snowflake and other fractal curves. There are also Jordan curves of positive area (first constructed by Osgood in 1903).

In fact, as it is also explained in the Wikipedia article that you linked, the problem is actually solved for "well-behaved" curves, such as convex curves or piecewise analytic curves, i.e. for objects that are close to our intuitive notion of "continuous closed loop".

A possible strategy to solve the problem in the general case is to try to approximate your Jordan curve by using well-behaved curves, for which we know that the conjecture is true, and then pass to the limit. The technical difficulty with this approach is that a limit of squares is not necessarily a square, but it can be a single point (i.e., a square "of side length 0"), so it is not clear whether it is always possible to find an approximating sequences of curves that works.

Let me elaborate on Sam Hopkins' comment.

The main reason that makes this and other problems on continuous curves so hard is that a "simple closed curve" or "Jordan curve", i.e. a non-self-intersecting continuous loop in the plane, can be a horrible object, for instance a nowhere differentiable curve such as the Koch snowflake and other fractal curves. There are also Jordan curves of positive area (first constructed by Osgood in 1903).

In fact, as it is also explained in the Wikipedia article that you linked, the problem is actually solved for "well-behaved" curves, such as convex curves or piecewise analytic curves, i.e. for objects that are close to our intuitive notion of "closed loop".

A possible strategy to solve the problem in the general case is to try to approximate your Jordan curve by using well behaved curves, for which we know that the result holds true, and then pass to the limit. The technical difficulty with this approach is that a limit of squares is not necessarily a square, but it can be a single point (i.e., a square "of side length 0"), so it is not clear whether it is always possible to take an approximating sequences of curves that works.

Let me elaborate on Sam Hopkins' comment.

The main reason that makes this and other problems on continuous curves so hard is that a "simple closed curve" or "Jordan curve", i.e. a non-self-intersecting continuous loop in the plane, can be a horrible object, for instance a nowhere differentiable curve such as the Koch snowflake and other fractal curves. There are also Jordan curves of positive area (first constructed by Osgood in 1903).

In fact, as it is also explained in the Wikipedia article that you linked, the problem is actually solved for "well-behaved" curves, such as convex curves or piecewise analytic curves, i.e. for objects that are close to our intuitive notion of "continuous closed loop".

A possible strategy to solve the problem in the general case is to try to approximate your Jordan curve by using well behaved curves, for which we know that the result holds true, and then pass to the limit. The technical difficulty with this approach is that a limit of squares is not necessarily a square, but it can be a single point (i.e., a square "of side length 0"), so it is not clear whether it is always possible to take an approximating sequences of curves that works.

Let me elaborate on Sam's comment.

The reason is that a "simple closed curve" or "Jordan curve", i.e. a non-self-intersecting continuous loop in the plane, can be a horrible object, for instance a nowhere differentiable curve such as the Koch snowflake and other fractal curves. There are also Jordan curves of positive area (constructed by Osgood in 1903).

In fact, as it is also explained in the Wikipedia article that you linked, the problem is solved for "well-behaved" curves, such as convex curves or piecewise analytic curves, that are objects close to our intuitive notion of "closed loop".

A possible strategy to solve the problem in the general case is to try to approximate your Jordan curve by using well behaved curves, for which we know that the result holds true. The technical problem with this approach is that a limit of squares is not necessarily a square, but it can be a point (i.e., a square "of side length 0").

Let me elaborate on Sam Hopkins' comment.

The main reason that makes this and other problems on continuous curves so hard is that a "simple closed curve" or "Jordan curve", i.e. a non-self-intersecting continuous loop in the plane, can be a horrible object, for instance a nowhere differentiable curve such as the Koch snowflake and other fractal curves. There are also Jordan curves of positive area (first constructed by Osgood in 1903).

In fact, as it is also explained in the Wikipedia article that you linked, the problem is actually solved for "well-behaved" curves, such as convex curves or piecewise analytic curves, i.e. for objects that are close to our intuitive notion of "closed loop".

A possible strategy to solve the problem in the general case is to try to approximate your Jordan curve by using well behaved curves, for which we know that the result holds true, and then pass to the limit. The technical difficulty with this approach is that a limit of squares is not necessarily a square, but it can be a single point (i.e., a square "of side length 0"), so it is not clear whether it is always possible to take an approximating sequences of curves that works.