2024-11-21

Recall our counterexample to non-equivalence:

$$\mathcal{U}=\{B_1((0,0)),B_1((0,1)),B_1((1,0)),B_1((1,1))\}.$$

We seek the exact birth times of simplices. We need to find the area of intersection of 2, 3 and 4 balls.

Two balls (aligned):

$$\frac{4\pi -3\sqrt{3}}{6}\approx 1.288$$

Two balls (diagonal):

$$\frac{\pi }{2}-1\approx 0.571$$

Three balls:

$$\frac{5\pi -6\sqrt{3}}{12}\approx 0.443$$

Four balls:

$$\frac{\pi +3-3\sqrt{3}}{3}\approx 0.321$$

Now we need to find the area of union of 2,3, and 4 balls.

Two balls (aligned):

$$2\mu (\text{ball})-\mu (\cap \text{2}\text{balls})=2\pi -\frac{4\pi -3\sqrt{3}}{6}=\frac{8\pi +3\sqrt{3}}{6}\approx 5.054$$

Two balls (diagonal):

$$2\mu (\text{ball})-\mu (\cap \text{2balls})=2\pi -\frac{\pi }{2}+1=\frac{3\pi }{2}+1\approx 0.571$$

Three balls:

$$\begin{array}{rl}3\mu (\text{ball})-2\mu (\cap \text{balls aligned})-\mu (\cap \text{balls diagonal})+\mu (\cap 3\text{balls})& =3\pi -\frac{4\pi -3\sqrt{3}}{3}-\frac{\pi }{2}+1+\frac{5\pi -6\sqrt{3}}{12}\\ & =\frac{19\pi +6\sqrt{3}+12}{12}\\ & \approx 6.84\end{array}$$

Four balls:

(https://math.stackexchange.com/questions/4115743/derive-an-expression-to-find-the-cardinality-of-the-union-of-4-sets)

$$\begin{array}{r}4\mu (B)-\mu (B_{0,0}\cap B_{0,1})-\mu (B_{0,0}\cap B_{1,0})-\mu (B_{0,0}\cap B_{1,1})\\ -\mu (B_{0,1}\cap B_{1,0})-\mu (B_{0,1}\cap B_{1,1})-\mu (B_{1,0}\cap B_{1,1})\\ +\mu (B_{0,0}\cap B_{0,1}\cap B_{1,0})+\mu (B_{0,0}\cap B_{0,1}\cap B_{1,1})\\ +\mu (B_{0,0}\cap B_{1,0}\cap B_{1,1})+\mu (B_{0,1}\cap B_{1,0}\cap B_{1,1})\\ -\mu (B_{0,0}\cap B_{0,1}\cap B_{1,0}\cap B_{1,1})\end{array}$$

which simplifies to

$$4\mu (B)-4\mu (B_{0,0}\cap B_{0,1})-2\mu (B_{0,0}\cap B_{1,1})+4\mu (\text{3 balls})-\mu (\text{4 balls})$$

i.e.,

$$\begin{array}{rl}& 4\pi -4(\frac{4\pi -3\sqrt{3}}{6})-2(\frac{\pi }{2}-1)+4(\frac{5\pi -6\sqrt{3}}{12})-\frac{\pi +3-3\sqrt{3}}{3}\\ & =4\pi -\frac{16\pi -12\sqrt{3}}{6}-\pi +2+\frac{20\pi -24\sqrt{3}}{12}-\frac{\pi +3-3\sqrt{3}}{3}\\ & =\frac{36\pi }{12}-\frac{32\pi -24\sqrt{3}}{12}+\frac{20\pi -24\sqrt{3}}{12}-\frac{4\pi +12-12\sqrt{3}}{12}\\ & =\frac{19\pi +12\sqrt{3}+12}{12}\\ & \approx 7.706\end{array}$$

Now we can compute the birth times:

1-simplices (aligned):

$$\begin{array}{rl}1-\frac{\mu (\cap )}{\mu (\cup )}& =1-\frac{\frac{4\pi -3\sqrt{3}}{6}}{\frac{8\pi +3\sqrt{3}}{6}}\\ & =1-\frac{4\pi -3\sqrt{3}}{8\pi +3\sqrt{3}}\\ & =\frac{4\pi +6\sqrt{3}}{8\pi +3\sqrt{3}}\\ & \approx 0.756\end{array}$$

1-simplices (diagonal):

$$\begin{array}{rl}1-\frac{\mu (\cap )}{\mu (\cup )}& =1-\frac{\pi /2-1}{3\pi /2+1}\\ & =1-\frac{\pi -2}{3\pi +2}\\ & =\frac{2\pi +4}{3\pi +2}\\ & \approx \mathrm{0.9.}\end{array}$$

2-simplices:

$$\begin{array}{rl}1-\frac{\mu (\cap )}{\mu (\cup )}& =1-\frac{\frac{5\pi -6\sqrt{3}}{12}}{\frac{19\pi +6\sqrt{3}+12}{12}}\\ & =1-\frac{5\pi -6\sqrt{3}}{19\pi +6\sqrt{3}+12}\\ & =\frac{14\pi +12\sqrt{3}+12}{19\pi +6\sqrt{3}+12}\\ & \approx 0.935\end{array}$$

3-simplex:

$$\begin{array}{rl}1-\frac{\mu (\cap )}{\mu (\cup )}& =1-\frac{\frac{\pi +3-3\sqrt{3}}{3}}{\frac{19\pi +12\sqrt{3}+12}{12}}\\ & =1-\frac{4\pi +12-12\sqrt{3}}{19\pi +12\sqrt{3}+12}\\ & =\frac{15\pi +24\sqrt{3}}{19\pi +12\sqrt{3}+12}\\ & \approx 0.959\end{array}$$