Let $d$ be a norm-based metric in $\mathbb{R}^2$. We are given a $d$-ball with radius 1, and we would like to cover it with objects with diameter 1. How many objects are needed?
In the $\ell_1$ metric, 4 are sufficient (and probably necessary):
Similarly, in the $\ell_\infty$ metric, 4 are sufficient (and probably necessary):
In the $\ell_2$ metric, 7 are sufficient, according to this answer by Joseph O'Rourke:
For covering with disks, 7 are also necessary (it cannot be covered by 6 disks), but maybe it can be covered by other objects with diameter 1.
QUESTION: is there a finite integer $C$ such that, for every metric $d$, every $d$-ball of radius 1 can be covered by $C$ objects of diameter 1?
NOTE: the following related questions are interesting too:
is there a finite integer $B$ such that, for every metric $d$, every $d$-ball of radius 1 can be covered by $B$ $d$-balls of diameter 1?
is there a finite integer $A$ such that, for every metric $d$, every $d$-ball of radius 1 can be covered by $A$ $d$-balls of radius $1/2$?
It is easy to see that $C\leq B\leq A$ (the latter follows from the fact that each $d$-ball of radius $1/2$ has diameter at most $1$ by the triangle equality).
EDIT 1: Anton Petrunin showed that $A$ is finite (and therefore $B$ and $C$ are finite too). A followup question is: what are tight upper bounds on these numbers?
I conjecture that $A\leq 16$: we can take the unit ball and apply an affinte transformation on it such that it contains a unit square, and is contained in a square of side-length at most 2 (I am not sure about the 2, but it holds for a triangle, and triangle seems a worst-case for all convex figures). Therefore, we can cover the unit ball by $4\times 4$ squares of side-length $1/2$; each of these is contained in a ball of radius $1/2$. Is the conjecture true? Is it possible to get tighter upper bounds on $A, B$ or $C$?
EDIT 2: in this paper we prove an upper bound $A\leq 25$, as well as motivate the search for tighter bounds: it provides a constant-factor approximation to an NP-hard matching problem.