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?

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    $\begingroup$ Are the copies allowed to be rotated? If so, one can get an ellipse with arbitrarily high kissing number. Otherwise, I imagine it would be six just as for circles. Gerhard "Usually Kisses One Convex Body" Paseman, 2019.11.20. $\endgroup$ – Gerhard Paseman Nov 21 '19 at 5:30
  • $\begingroup$ Yes. The copies can be rotated. Edited the question to that effect. As you point out, for unit A and large P, C is a 'long' region and the kissing number can be very large. The question is: among all those long convex regions with large kissing numbers, whether it is the ellipse that has the least kissing number. $\endgroup$ – Nandakumar R Nov 21 '19 at 8:25
  • $\begingroup$ Do you know a convex shape with kissing number $<6$? Maybe some (regular) $n$-gon? It is always $\ge 6$ for ellipses. $\endgroup$ – M. Winter Nov 21 '19 at 14:26
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    $\begingroup$ @M.W, Zhao and Xu, The kissing number of the regular pentagon, Disc. Math. 252 (2002) 293-298, establish that the kissing number of the regular pentagon is $6$. Together with earlier work they cite, this proves the kissing number of the regular $n$-gon is $6$ for all $n\ge5$. $\endgroup$ – Gerry Myerson Nov 22 '19 at 11:55
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    $\begingroup$ @M.Winter The construction of the optimal double lattice packing (not lattice packing, see doi.org/10.1007/BF02187800) implies six contact points with neighboring copies. Might not actually be the optimal packing without restricting to double lattices, but we are not interesting in density in this question, just contact number, no? Also, might not be highest possible contact number, but it's a lower bound, so no convex shape can have kissing number < 6. $\endgroup$ – Yoav Kallus Nov 22 '19 at 16:14

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