Convex Regions and 'Kissing Numbers'

This post continues a thread from this post: https://nandacumar.blogspot.com/2019/01/a-bit-more-on-kissing-numbers.html

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.

Question: Given values A and P, which is the convex region C with area = A and 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 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?