I added the related pages from part 3 of the book: combinatorial geometry by János Pach,Pankaj K.Agarwal (1995) (which is not available on net so I added them as pictures).

A. Prove that one can always find a packing $Ç$ of the plane with congruent copies of a convex disc $C$ whose density $d(Ç,R^2)$ exists and is equal to $\delta(C)$. Similarly show that there is a Lattice packing $Ç$ with $d(Ç,R^2)=\delta_L(C)$.

B.construct a packing $Ç$ of unit discs and two convex discs $P,P'$ such that $lim_{r\to \infty} d(Ç,P(r)) \neq lim_{r\to \infty} d(Ç,P'(r))$.

Hints: Part A:To prove the first statement, construct a packing $C_n$ of congruent copies of $C$ in the disc $D(n)$ with density $d(C_n,D(n))\geq \delta(C)-O(1)/n$

For every n choose a subsequence $n_1,n_2,...$ such that $C_{n_i}$ converges when restricted to $D(1)$ as $i\to \infty$ . From this choose a subsequence for which $C_n$ converges when restricted to $D(2)$ and so on. Show that the limit packing meets the requirements.

For part B: Let $Ç$ be the densest packing of unit discs in a cone. Choose $P$ and $P'$ to be circular disc and a regular triangle respectively.

Can you kindly help me to understand this (I'm a beginner in this field).

1.(https://i.stack.imgur.com/2k1mz.jpg)

2.(https://i.stack.imgur.com/ZEdCp.jpg)

3.(https://i.stack.imgur.com/fhhvM.jpg)

4.(https://i.stack.imgur.com/WIVsz.jpg)

5.(https://i.stack.imgur.com/B2O4m.jpg)

6.(https://i.stack.imgur.com/ZuXvX.jpg)

Many thanks.