Given a 2D convex region $C$, let us define its kissing number $K_0$ to be the largest possible number of copies of $C$ that can be arranged around a central copy of $C$ (call this $C_0$) and touching $C_0$. Note that the surrounding copies can have any orientation.

Question: Given values $A$ and $P$, which is the convex region $C$ with ${\rm area} = A$ and ${\rm perimeter} = P$ and with **least** $K_0$? Is it always an ellipse?

Note: One can also define '$K_1$' to be *the largest number of copies of $C$ that can be arranged around a central $C_0$ such that the copies touch either $C_0$ or a copy that touches $C_0$*. So one can ask the shape of $C$ with specified $A$ and $P$ and with **least** $K_1$. And these questions have natural higher dimensional analogs.

Further question: Given any convex $C$, we need to find its $K_1$. IOW, we need to find an arrangement of copies of $C$ around a central $C_0$ such that they all touch either $C_0$ or a copy that touches $C_0$. Now, is it sufficient to first arrange $K_0$ copies all kissing $C_0$ and then to arrange maximum number of copies all touching at least one of these $K_0$ copies? Are there $C$'s where such a 'greedy' approach fails?