Polya's orchard problem asks for which radius $\rho$ of trees at each lattice point within a distance $R$ of the origin block all lines of sight to the exterior of the orchard.

It has been established that rays to infinity are completely blocked iff
$\rho \ge 1/\sqrt{R^2 + 1}$, when $R$ is an integer.
(T.T. Allen, "Polya's orchard problem,"
*The American Mathematical Monthly*
93(2): 98-104 (1986).)
The above shows a quarter of an orchard with $R=6$, $\rho=1/\sqrt{37}=0.164$,
and some random rays.

I am wondering if disks are the most efficient blockers in terms of area. More precisely:

For a given $R$, is there a centrally symmetric convex body $K$ of area less than $\pi \rho^2$ which when translated to all lattice points within distance $R$ of the origin, block all rays from the origin to the outside?

My guess is that the answer is *Yes*, in which case it would
be interesting to know the shape of the area-optimal blockers.
In particular, are they polygons?
The same question may be posed in $\mathbb{R}^d$: are they polytopes?

**Edit**. Here is the chord construction for $R=2$ from the first paragraph of Douglas's construction, as I understand it:

improvementThis problem is kind of subtle. One could do this same cord construction allowing disks of different sizes at different points. Furthermore, the horizontal and vertical cords of length $\frac45$ can be replaced by segments of length $1$ going through the center of each circle. These are longer but also more central and the area turns out to be smaller. I added a picture of this to my answer below. $\endgroup$more improvementAnd even that is not optimal. I think that squares of side $\frac23$ are. I added yet another picture. $\endgroup$