4 added 90 characters in body

Let $S$ be a closed convex surface, the boundary of a compact convex body in $\mathbb{R}^3$. I am interested in whether there are conditions on its shape that ensure that it supports a long, simple (non-self-crossing) geodesic. The length of a geodesic for my purposes is the longest distance you can travel along the geodesic before returning to your starting point. Some condition is necessary for the type of result I seek, for all the geodesics on a sphere have the same length.

Define the elongation $L$ of $S$ as the largest height to diameter ratio, $h/d$, of a cylinder of height $h$ and diameter $d$ in which $S$ is tightly inscribed. By tightly inscribed I mean that $S$ touches the top, bottom, and sides of the cylinder in such a manner that neither the height nor diameter can be reduced.

I could use a theorem of this type:

If $S$ has elongation $L \ge k$, then there is a simple geodesic on $S$ of length $\ge f(k)$, where $f(k)$ is some increasing function of $k$, e.g., $c k$ for a constant $c > 0$.

Perhaps such a theorem cannot exist. Or maybe a theorem of this ilk exists, but only with certain smoothness assumptions? There are always at least three simple closed geodesics on $S$, by a theorem of Lyusternik and Schnirelmann, but perhaps they might all be short?

For an ellipsoid, the three simple closed geodesics follow the major and minor axes, and the longest of those satisfies the type of relationship I seek. (Elongation could as well be defined in terms of an enclosing ellipsoid rather than cylinder.) And a cylindrical $S$ supports a long spiral geodesic:

Such spirals are exactly the type of geodesic I seek. Thanks for any ideas or pointers!

Edit. This may not add much, but here is how I view a long geodesic on a cylinder: starting at $a$, crossing the bottom in a segment $x x'$, crossing the top in $y y'$, and stopping at $b$ just before it is about to cross itself.

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Let $S$ be a closed convex surface, the boundary of a compact convex body in $\mathbb{R}^3$. I am interested in whether there are conditions on its shape that ensure that it supports a long, simple (non-self-crossing) geodesic. The length of a geodesic for my purposes is the longest distance you can travel along the geodesic before returning to your starting point. Some condition is necessary for the type of result I seek, for all the geodesics on a sphere have the same length.

Define the elongation $L$ of $S$ as the largest height to diameter ratio, $h/d$, of a cylinder of height $h$ and diameter $d$ in which $S$ is tightly inscribed. By tightly inscribed I mean that $S$ touches the top, bottom, and sides of the cylinder in such a manner that neither the height nor diameter can be reduced.

I could use a theorem of this type:

If $S$ has elongation $L \ge k$, then there is a simple geodesic on $S$ of length $\ge f(k)$, where $f(k)$ is some increasing function of $k$, e.g., $c k$ for a constant $c > 0$.

Perhaps such a theorem cannot exist. Or maybe a theorem of this ilk exists, but only with certain smoothness assumptions? There are always at least three simple closed geodesics on $S$, by a theorem of Lyusternik and Schnirelmann, but perhaps they might all be short?

For an ellipsoid, the three simple closed geodesics follow the major and minor axes, and the longest of those satisfies the type of relationship I seek. (Elongation could as well be defined in terms of an enclosing ellipsoid rather than cylinder.) And a cylindrical $S$ supports a long spiral geodesic:

Such spirals are exactly the type of geodesic I seek. Thanks for any ideas or pointers!

Edit. This may not add much, but here is how I view a long geodesic on a cylinder: starting at $a$, crossing the bottom in a segment $x x'$, crossing the top in $y y'$, and stopping at $b$ just before it is about to cross itself.

2 Added geodesic on a cylinder.

Let $S$ be a closed convex surface, the boundary of a compact convex body in $\mathbb{R}^3$. I am interested in whether there are conditions on its shape that ensure that it supports a long, simple (non-self-crossing) geodesic. The length of a geodesic for my purposes is the longest distance you can travel along the geodesic before returning to your starting point. Some condition is necessary for the type of result I seek, for all the geodesics on a sphere have the same length.

Define the elongation $L$ of $S$ as the largest height to diameter ratio, $h/d$, of a cylinder of height $h$ and diameter $d$ in which $S$ is tightly inscribed. By tightly inscribed I mean that $S$ touches the top, bottom, and sides of the cylinder in such a manner that neither the height nor diameter can be reduced.

I could use a theorem of this type:

If $S$ has elongation $L \ge k$, then there is a simple geodesic on $S$ of length $\ge f(k)$, where $f(k)$ is some increasing function of $k$, e.g., $c k$ for a constant $c > 0$.

Perhaps such a theorem cannot exist. Or maybe a theorem of this ilk exists, but only with certain smoothness assumptions? There are always at least three simple closed geodesics on $S$, by a theorem of Lyusternik and Schnirelmann, but perhaps they might all be short?

For an ellipsoid, the three simple closed geodesics follow the major and minor axes, and the longest of those satisfies the type of relationship I seek. (Elongation could as well be defined in terms of an enclosing ellipsoid rather than cylinder.) And a cylindrical $S$ supports a long spiral geodesic:

Such spirals are exactly the type of geodesic I seek. Thanks for any ideas or pointers!

Edit. This may not add much, but here is how I view a long geodesic on a cylinder: starting at $a$, crossing the bottom in a segment $x x'$, crossing the top in $y y'$, and stopping at $b$ just before it is about to cross itself.

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