Rubber band domains "Rubber band" domains are defined below.
(i) Is this class of domains already named?
If so, what is the name?
(ii) What results about rubber
band domains are known/published?
This class of domains seems to be relevant
to the "Lion and Man" pursuit problem.
The crucial condition in the following
definition is (d).
Definition. A set $D$ will be called a rubber
band domain if it satisfies the following
conditions.
(i) $D$ is an open bounded connected subset
of $\mathbb R^3$. Its boundary is smooth.
(ii) "Every rubber band is contractible
to a point." More precisely, if K is
a subset of $D$, it is homeomorphic to
the unit circle $U$ and has a finite
length (i.e., $K$ is a rectifiable Jordan curve) 
then there exists a continuous function $M(x,t)$ 
defined on  $U \times [0,1]$ such that
(a) for every $t \in [0,1)$, $M(x,t)$ is
a homeomorphism between $U$ and its range
(b) the range of $M(x,1$) is a single point in $D$
(c) for any $t \in [0,1]$, the range of $M(x, t)$
is a rectifiable Jordan curve
(d) if $s < t$ and $s,t$ belong to $[0,1]$
then the length of the range of $M(x,s)$
is not smaller than the length
of the range of $M(x,t)$.
 A: Any convex polyhedron is a rubber-band domain if you drop the requirement that $D$ is open.
To see this, take a loop, and push it around until it becomes a geodesic. Now translate the entire loop in a direction orthogonal to the loop, keeping the length of the loop constant, until it bumps into a vertex. It won't unexpectedly intersect itself while you are doing this. Now if you push a part of the loop past that vertex, the loop decreases in length since the polyhedron is convex. Now push it around again until it becomes a geodesic, and repeat. There are only finitely many possible lengths for a closed geodesic that doesn't intersect itself, so this process eventually finishes.
More generally, I think any convex surface should be a rubber band domain.
Edit: I was thinking about this problem today, and it finally hit me!
If $D$ is the complement of an object (intersected with a large ball containing the surface), then this is exactly the condition that there is no knot that can't be slipped off the object. So by the answers to the question I posed here, we have the results that:
(1) The complement of a ball (intersected with the interior of a larger ball) is a rubber band domain.
(2) The complement of an equilateral triangle (intersected with the interior of a large ball) is not a rubber band domain.
