Suppose you have two compact convex sets $A,B \subset \mathbb R^2$. Their interiors intersect, but neither is contained in the other. Do the boundaries intersect in at least two distinct points?
Obviously in general the intersection might be a singleton. Take two discs tangent to each other:
But in this case the interiors are disjoint, so it doesn't count. I am interested in the case where the interiors intersect. Like this:
Here you can see the boundaries intersect exactly twice. In fact we can get four intersections easily enough:
Is it easy to prove there are always at least 2 distinct intersections?
There is some point $p_1$ on the boundary of $A$ is in the interior of $B$, and some point $p_2$ on the boundary of $A$ that is not in $B$. As you go along the boundary of $A$ from $p_1$ to $p_2$, you must hit the boundary of $B$ at some point. But there are two ways (disjoint except for the endpoints) to go from $p_1$ to $p_2$ on the boundary, so two such points.