# Probabilities of Four Points Being in Convex/Deltoid Configurations

Question:

what is the probability that four distinct points in general position in the Euclidean plane are in convex configuration, depending on the number of leaf nodes in their Minimum Spanning Tree (MST)?

Put differently, what are the probabilities for the following situations:

• the points are in convex configuration and
• the MST has two leaf nodes?
• the MST has three leaf nodes?
• the points are in deltoid configuration
• the MST has two leaf nodes?
• the MST has three leaf nodes?

I assume that a quadruplet of points has a higher probability of being in convex configuration if its MST has two leaf nodes than if it has three, but that that probability is less than 1.

Denote by $$P_1,P_2,P_3,P_4$$ the $$4$$ random points and by $$p_{12}$$ the probability that the points $$P_3,P_4$$ lie on the same side of the line determined by $$P_1,P_2$$. Then, as shown by Renyi and Sulanke we have $$\bar{v}=\binom{4}{2}p_{12}= 6p_{12}.$$ Given two points $$P,Q$$ we denote by $$L(P,Q)$$ the line determined by $$P,Q$$ and by $$a(P,Q)$$ the area of the part of the disk that lies on the same side of $$L(P,Q)$$ as the center of the disk. Set $$\bar{a}(P,Q)=\frac{1}{\pi} a(P,Q).$$ Then $$p_{12}= \frac{1}{\pi^2}\int_{\bD^2}\Big(\; \bar{a}(P,Q)^2+(1-\bar{a}(P,Q))^2\;\Big)dPdQ$$ $$=\int_{\bD^2}\Big(\; 2\bar{a}(P,Q)^2-2\bar{a}(P,Q)+1\;\Big) dPdQ$$ $$=1-2\int_{\bD^2} \bar{a}(P,Q)\big(\; 1-\bar{a}(P,Q)\;\big) dPdQ.$$ The line $$L(P,Q)$$ has a unique equation of the form $$x\cos \theta+y\sin\theta =p,\;\;p\in[0,1], \;\;\theta\in[0,2\pi].$$ We denote by $$L_{\theta,p}$$ the line described by such an equation. The chord cout-ut by the disk on this line has length $$\ell(p)=2\sqrt{1-p^2}.$$ For such a line we have $$a(P,Q)=\frac{\pi}{2}+2\int_0^p \sqrt{1-x^2} dx =\frac{\pi}{2} +p\sqrt{1-p^2} +\arcsin(p).$$ We set $$\alpha(p):=\frac{1}{2}+ \frac{1}{\pi} \big(\; p\sqrt{1-p^2} +\arcsin(p)\;\big)=\bar{a}(P,Q).$$ As explained in the equality (5) in here we have $$\int_{\bD^2} \bar{a}(P,Q)\big(\; 1-\bar{a}(P,Q)\;\big) dPdQ=\frac{1}{3}\int_0^{2\pi} \int_0^1\alpha(p)(1-\alpha(p)) \ell(p)^3 dp d\theta,$$ $$=\frac{2\pi}{3} \int_0^1 \alpha(p)(1-\alpha(p)) \ell(p)^3 dp.$$ This integral can be computed/approximated using a computer algebra system such as MAPLE or MATHEMATICA.