In the case of a closed curve my first wild (but educated :-) guess would be: connect the following $\ 8\ $points (vertices of a cube but also belonging to a sphere) in the given cyclic order (of a maximal shift register):

$$(-a\ -\!a\ -\!a)\quad(-a\ -\!a\ +\!a)\quad(-a\ +\!a\ +\!a)\quad(-a\ +\!a\ -\!a)$$ $$(+a\ +\!a\ -\!a)\quad(+a\ +\!a\ +a)\quad(+a\ -\!a\ +\!a)\quad(+a\ -\!a\ -\!a) $$

where $\ a\ $ is such that the length of the large arc which connects the consecutive points on the sphere which contains these $\ 8\ $ points is $\ \frac18$.

EDIT Shooting from the hip cannot be that precise. Thus two related questions may help some:

1. What is a maximal volume of a convex hull of a closed curve contained in a sphere?

2. Is there such a curve but spherical which encloses a larger volume (i.e. its convex hall would be larger than for any closed curve contained in a sphere)?

In the case of a closed curve my first wild (but educated :-) guess would be: connect the following $\ 8\ $points (vertices of a cube but also belonging to a sphere) in the given cyclic order (of a maximal shift register):

$$(-a\ -\!a\ -\!a)\quad(-a\ -\!a\ +\!a)\quad(-a\ +\!a\ +\!a)\quad(-a\ +\!a\ -\!a)$$ $$(+a\ +\!a\ -\!a)\quad(+a\ +\!a\ +a)\quad(+a\ -\!a\ +\!a)\quad(+a\ -\!a\ -\!a) $$

where $\ a\ $ is such that the length of the large arc which connects the consecutive points on the sphere which contains these $\ 8\ $ points is $\ \frac18$.

EDIT Shooting from the hip cannot be that precise. Thus two related questions may help some:

1. What is a maximal volume of a convex hull of a closed curve contained in a sphere? (Then the respective radius would be of interest).

2. Is there such a curve but not spherical which encloses a larger volume (i.e. its convex hall would be larger than for any closed curve contained in a sphere)?

In the case of a closed curve my first wild (but educated :-) guess would be: connect the following $\ 8\ $points (vertices of a cube but also belonging to a sphere) in the given cyclic order (of a maximal shift register):

$$(-a\ -\!a\ -\!a)\quad(-a\ -\!a\ +\!a)\quad(-a\ +\!a\ +\!a)\quad(-a\ +\!a\ -\!a)$$ $$(+a\ +\!a\ -\!a)\quad(+a\ +\!a\ +a)\quad(+a\ -\!a\ +\!a)\quad(+a\ -\!a\ -\!a) $$

where $\ a\ $ is such that the length of the large arc which connects the consecutive points on the sphere which contains these $\ 8\ $ points is $\ \frac18$.

In the case of a closed curve my first wild (but educated :-) guess would be: connect the following $\ 8\ $points (vertices of a cube but also belonging to a sphere) in the given cyclic order (of a maximal shift register):

$$(-a\ -\!a\ -\!a)\quad(-a\ -\!a\ +\!a)\quad(-a\ +\!a\ +\!a)\quad(-a\ +\!a\ -\!a)$$ $$(+a\ +\!a\ -\!a)\quad(+a\ +\!a\ +a)\quad(+a\ -\!a\ +\!a)\quad(+a\ -\!a\ -\!a) $$

where $\ a\ $ is such that the length of the large arc which connects the consecutive points on the sphere which contains these $\ 8\ $ points is $\ \frac18$.

EDIT Shooting from the hip cannot be that precise. Thus two related questions may help some:

1. What is a maximal volume of a convex hull of a closed curve contained in a sphere?

2. Is there such a curve but spherical which encloses a larger volume (i.e. its convex hall would be larger than for any closed curve contained in a sphere)?