2 added 245 characters in body

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 Ls, 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 Ls, 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 Ls, 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 $$iv=\frac{1}{4}$$.)

1

Shape that balances isoperimetric ratio with invisibility

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 Ls, 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 $$P(a)$$.