Given two potatoes, prove that there is a loop of wire which fits around both

Question

This is a classic problem in geometric continuity and I want to see if there are some solutions other than the one I'm thinking of:

Two potatoes are given. Prove that there exists a closed loop of wire (fixed length) that fits tightly around each potato.

A more mathematical way of saying this is: we are given two closed, non intersecting 3-dimensional surfaces $S_1$ and $S_2$. Prove that there is some closed, non-intersecting path $p_1$ on $S_1$, and some closed, non-intersecting path $p_2$ on $S_2$, such that $p_1$ and $p_2$ are identical in shape.

Answer 1

Take your two potatoes, and intersect them. They'll intersect in a closed curve, which is your wire. (If they intersect in multiple closed curves, just pick one of them.)