I have two solid two-dimensional shapes on a plane, $S_1$ and $S_2$, with boundaries $B_1$ and $B_2$. If the boundaries of the two shapes are allowed to touch, but no points internal to the boundaries are allowed to intersect, is it true that the boundaries of $S_1$ and $S_2$ can touch at only a single point on a plane (allowing for rotation, translation, reflection, etc. of the shapes) if and only if $S_1$ and $S_2$ are strictly convex? How does this generalize to higher dimensions?
Update in response to WimC's comment: Can we say that $S_1$ and/or $S_2$ are strictly convex if and only if their boundaries can be touched together at most at a single point?
Update 2: Perhaps we can rescue the original statement of this problem, requiring $S_1$ and $S_2$ to both be strictly convex, if we require that there should only ever be a single point of intersection for arbitrary dilations of either shape?
Update 3: In response to Rahul's comment, the converse statement of want to show here is that, provided neither $S_1$ or $S_2$ are strictly convex, then there MUST exist some configuration of the shapes that allows for multiple simultaneous contacts along their boundaries. We know that if both shapes are strictly convex then there can only be one point of contact, and we've discussed conditions where one point of contact can exist when only one shape is strictly convex, but it seems more challenging to show that one shape being strictly convex is both necessary and sufficient for forbidding multiple points of simultaneous contact.
For weakly convex functions, i.e. convex hulls etc. with straight lines in their boundaries, can we make a similar statement about the number of edges that can touch?
Please note that this is a (hopefully) more focused version of an earlier question, now removed, which asked for the properties of $B_1$ and $B_2$ that guaranteed some strict upperbound for the number of points $k$ where $B_1$ and $B_2$ can touch.