I seek a two-dimensional shapes $S$, bounded by a Jordan curve,
that optimally balances its isoperimetric ratio $r(S)$
against what I call its invisibility index $iv(S)$.
Define the *isoperimetric ratio* $r(S)$ of $S$ to be
$4 \pi A / L^2$, where $A$ is the area of $S$ and $L$ its
perimeter. This ratio is in $(0,1]$
and achieves $1$ for $S$ a disk.
See, e.g., the
Wikipedia article on the isoperimetric inequality.
Define *invisibility index* $iv(S)$
to be the probability that that a pair $(x,y)$ of random points
in $S$
(chosen uniformly and independently)
are invisible to one another in the sense that
the segment $xy$ includes a point strictly exterior to $S$.
($iv(S)$ is $1$ minus the
Beer convexity of $S$.)

**Q**. What shape (or shapes) $S$ maximize the product
$P(S) = r(S) \cdot iv(S)$?

If $S$ is a disk, $r(S)=1$ and $iv(S)=0$ so $P(S)=0$.
If $S$ is a thin spiral, then $r(S)$ approaches $0$
and $iv(S)$ approaches $1$ so $P(S)$ approaches $0$.
In between, $P(S) > 0$.

I've computed $P(S)$ for the very narrow class of
symmetric `L`

s, unit squares with a square removed
from one corner, as illustrated below.

^{
Two symmetric Ls with different parameters $a$. Origin at lowerleft corner.
}

These shapes are determined by one parameter

$a$ as illustrated.
Among this class of shapes, it appears that the maximum
product

$P(S)$ is achieved when

$a \approx \frac{1}{4}$,
the left shape above. Plots

$r(\,)$,

$iv(\,)$, and

$P(\,)$
are shown below.
The isoperimetric ratio for a square is

$r(1) = \pi/4 \approx 0.79$.

^{
Red: $r(a)$. Blue: $iv(a)$. Green: Product $P(a)$.
}

**Update**. Seems like Gerhard Paseman's figure-8
, with $r=\frac{1}{2}$, $iv=\frac{1}{2}$, $P=\frac{1}{4}$,
is the extreme shape. (In comments I mistakenly said $iv=\frac{1}{4}$.)

I seek a two-dimensional shapes $S$, bounded by a Jordan curve,
that optimally balances its isoperimetric ratio $r(S)$
against what I call its invisibility index $iv(S)$.
Define the *isoperimetric ratio* $r(S)$ of $S$ to be
$4 \pi A / L^2$, where $A$ is the area of $S$ and $L$ its
perimeter. This ratio is in $(0,1]$
and achieves $1$ for $S$ a disk.
See, e.g., the
Wikipedia article on the isoperimetric inequality.
Define *invisibility index* $iv(S)$
to be the probability that that a pair $(x,y)$ of random points
in $S$
(chosen uniformly and independently)
are invisible to one another in the sense that
the segment $xy$ includes a point strictly exterior to $S$.
($iv(S)$ is $1$ minus the
Beer convexity of $S$.)

**Q**. What shape (or shapes) $S$ maximize the product
$P(S) = r(S) \cdot iv(S)$?

If $S$ is a disk, $r(S)=1$ and $iv(S)=0$ so $P(S)=0$.
If $S$ is a thin spiral, then $r(S)$ approaches $0$
and $iv(S)$ approaches $1$ so $P(S)$ approaches $0$.
In between, $P(S) > 0$.

I've computed $P(S)$ for the very narrow class of
symmetric `L`

s, unit squares with a square removed
from one corner, as illustrated below.

^{
Two symmetric Ls with different parameters $a$. Origin at lowerleft corner.
}

These shapes are determined by one parameter

$a$ as illustrated.
Among this class of shapes, it appears that the maximum
product

$P(S)$ is achieved when

$a \approx \frac{1}{4}$,
the left shape above. Plots

$r(\,)$,

$iv(\,)$, and

$P(\,)$
are shown below.
The isoperimetric ratio for a square is

$r(1) = \pi/4 \approx 0.79$.

^{
Red: $r(a)$. Blue: $iv(a)$. Green: Product $P(a)$.
}

I seek a two-dimensional shapes $S$, bounded by a Jordan curve,
that optimally balances its isoperimetric ratio $r(S)$
against what I call its invisibility index $iv(S)$.
Define the *isoperimetric ratio* $r(S)$ of $S$ to be
$4 \pi A / L^2$, where $A$ is the area of $S$ and $L$ its
perimeter. This ratio is in $(0,1]$
and achieves $1$ for $S$ a disk.
See, e.g., the
Wikipedia article on the isoperimetric inequality.
Define *invisibility index* $iv(S)$
to be the probability that that a pair $(x,y)$ of random points
in $S$
(chosen uniformly and independently)
are invisible to one another in the sense that
the segment $xy$ includes a point strictly exterior to $S$.
($iv(S)$ is $1$ minus the
Beer convexity of $S$.)

**Q**. What shape (or shapes) $S$ maximize the product
$P(S) = r(S) \cdot iv(S)$?

If $S$ is a disk, $r(S)=1$ and $iv(S)=0$ so $P(S)=0$.
If $S$ is a thin spiral, then $r(S)$ approaches $0$
and $iv(S)$ approaches $1$ so $P(S)$ approaches $0$.
In between, $P(S) > 0$.

I've computed $P(S)$ for the very narrow class of
symmetric `L`

s, unit squares with a square removed
from one corner, as illustrated below.

^{
Two symmetric Ls with different parameters $a$. Origin at lowerleft corner.
}

These shapes are determined by one parameter

$a$ as illustrated.
Among this class of shapes, it appears that the maximum
product

$P(S)$ is achieved when

$a \approx \frac{1}{4}$,
the left shape above. Plots

$r(\,)$,

$iv(\,)$, and

$P(\,)$
are shown below.
The isoperimetric ratio for a square is

$r(1) = \pi/4 \approx 0.79$.

^{
Red: $r(a)$. Blue: $iv(a)$. Green: Product $P(a)$.
}

**Update**. Seems like Gerhard Paseman's figure-8
, with $r=\frac{1}{2}$, $iv=\frac{1}{2}$, $P=\frac{1}{4}$,
is the extreme shape. (In comments I mistakenly said $iv=\frac{1}{4}$.)