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I don't know if you already thought of this, but your question is related to a result of Balacheff and Sabourau (an electronic version of their paper is in Stéphane's web page). Without going into the technical definition of the diastole over $1$-cycles, their result basically says that there exists a constant $C$ (which, unfortunately, is very large) so that given a Riemannian metric on the sphere with area $4\pi$, there exists a Morse function such that the length of any of its level curves is less than $C$. This is the best that is known in this direction.

On the other hand, a related conjecture which has perhaps a shot at being true runs as follows:

Conjecture. If the area of a Riemannian two-sphere is $4\pi$, there exists a closed geodesic that is regular homotopic to a figure 8 and whose length is less than or equal to $4\pi$.

Note that twice an equator in a round sphere is regular homotopic to a figure 8.

In fact, the conjecture should be for all (reversible and non-reversible) Finsler metrics and equality should hold only for Zoll metrics. The reason is that this conjecture is a consequence of (a reasonable extension of) the Viterbo conjecture. Enough is known about this conjecture (Artstein-Avidan, Ostrover, Milman, Alvarez Paiva and Balacheff: see the first version of Contact geometry and isosystolic inequalities and the references therein) that an optimist may harbor some hope of its validity.

I don't know if you already thought of this, but your question is related to a result of Balacheff and Sabourau (an electronic version of their paper is in Stéphane's web page). Without going into the technical definition of the diastole over $1$-cycles, their result basically says that there exists a constant $C$ (which, unfortunately, is very large) so that given a Riemannian metric on the sphere with area $4\pi$, there exists a Morse function such that the length of any of its level curves is less than $C$. This is the best that is known in this direction.

On the other hand, a related conjecture which has perhaps a shot at being true runs as follows:

Conjecture. If the area of a Riemannian two-sphere is $4\pi$, there exists a closed geodesic that is regular homotopic to a figure 8 and whose length is less than or equal to $4\pi$.

Note that twice an equator in a round sphere is regular homotopic to a figure 8.

In fact, the conjecture should be for all (reversible and non-reversible) Finsler metrics and equality should hold only for Zoll metrics. The reason is that this conjecture is a consequence of (a reasonable extension of) the Viterbo conjecture. Enough is known about this conjecture (Artstein-Avidan, Ostrover, Milman, Alvarez Paiva and Balacheff: see the first version of Contact geometry and isosystolic inequalities and the references therein) that an optimist may harbor some hope of its validity.

I don't know if you already thought of this, but your question is related to a result of Balacheff and Sabourau (an electronic version of their paper is in Stéphane's web page). Without going into the technical definition of the diastole over $1$-cycles, their result basically says that there exists a constant $C > 0$ (which, unfortunately, is very small) so that given a Riemannian metric on the sphere with area $4\pi$, there exists a Morse function such that the length of any of its level curves is less than $C$. This is the best that is known in this direction.

On the other hand, a related conjecture which has perhaps a shot at being true runs as follows:

Conjecture. If the area of a Riemannian two-sphere is $4\pi$, there exists a closed geodesic that is regular homotopic to a figure 8 and whose length is less than or equal to $4\pi$.

Note that twice an equator in a round sphere is regular homotopic to a figure 8.

In fact, the conjecture should be for all (reversible and non-reversible) Finsler metrics and equality should hold only for Zoll metrics. The reason is that this conjecture is a consequence of (a reasonable extension of) the Viterbo conjecture. Enough is known about this conjecture (Artstein-Avidan, Ostrover, Milman, Alvarez Paiva and Balacheff: see the first version of Contact geometry and isosystolic inequalities and the references therein) that an optimist may harbor some hope of its validity.

I don't know if you already thought of this, but your question is related to a result of Balacheff and Sabourau (an electronic version of their paper is in Stéphane's web page). Without going into the technical definition of the diastole over $1$-cycles, their result basically says that there exists a constant $C$ (which, unfortunately, is very large) so that given a Riemannian metric on the sphere with area $4\pi$, there exists a Morse function such that the length of any of its level curves is less than $C$. This is the best that is known in this direction.

On the other hand, a related conjecture which has perhaps a shot at being true runs as follows:

Conjecture. If the area of a Riemannian two-sphere is $4\pi$, there exists a closed geodesic that is regular homotopic to a figure 8 and whose length is less than or equal to $4\pi$.

Note that twice an equator in a round sphere is regular homotopic to a figure 8.

In fact, the conjecture should be for all (reversible and non-reversible) Finsler metrics and equality should hold only for Zoll metrics. The reason is that this conjecture is a consequence of (a reasonable extension of) the Viterbo conjecture. Enough is known about this conjecture (Artstein-Avidan, Ostrover, Milman, Alvarez Paiva and Balacheff: see the first version of Contact geometry and isosystolic inequalities and the references therein) that an optimist may harbor some hope of its validity.

I don't know if you already thought of this, but your question is related to a result of Balacheff and Sabourau (an electronic version of their paper is in Stéphane's web page). Without going into the technical definition of the diastole over $1$-cycles, their result basically says that there exists a constant $C > 0$ (which, unfortunately, is very small) so that given a Riemannian metric on the sphere with area $4\pi$, there exists a Morse function such that the length of any of its level curves is less than $C$. This is the best that is known in this direction.

On the other hand, a related conjecture which has at least a shot at being true runs as follows:

Conjecture. If the area of a Riemannian two-sphere is $4\pi$, there exists a closed geodesic that is regular homotopic to a figure 8 and whose length is less than or equal to $4\pi$.

Note that twice an equator in a round sphere is regular homotopic to a figure 8.

In fact, the conjecture should be for all (reversible and non-reversible) Finsler metrics and equality should hold only for Zoll metrics. The reason is that this conjecture is a consequence of (a reasonable extension of) the Viterbo conjecture. Enough is known about this conjecture (Artstein-Avidan, Ostrover, Milman, Alvarez Paiva and Balacheff: see the first version of Contact geometry and isosystolic inequalities and the references therein) that an optimist may harbor some hope of its validity.

I don't know if you already thought of this, but your question is related to a result of Balacheff and Sabourau (an electronic version of their paper is in Stéphane's web page). Without going into the technical definition of the diastole over $1$-cycles, their result basically says that there exists a constant $C > 0$ (which, unfortunately, is very small) so that given a Riemannian metric on the sphere with area $4\pi$, there exists a Morse function such that the length of any of its level curves is less than $C$. This is the best that is known in this direction.

On the other hand, a related conjecture which has perhaps a shot at being true runs as follows:

Conjecture. If the area of a Riemannian two-sphere is $4\pi$, there exists a closed geodesic that is regular homotopic to a figure 8 and whose length is less than or equal to $4\pi$.

Note that twice an equator in a round sphere is regular homotopic to a figure 8.

In fact, the conjecture should be for all (reversible and non-reversible) Finsler metrics and equality should hold only for Zoll metrics. The reason is that this conjecture is a consequence of (a reasonable extension of) the Viterbo conjecture. Enough is known about this conjecture (Artstein-Avidan, Ostrover, Milman, Alvarez Paiva and Balacheff: see the first version of Contact geometry and isosystolic inequalities and the references therein) that an optimist may harbor some hope of its validity.