# Convex hull of prefix sum of $n$ ordered random points

Suppose we have $$n$$ ordered realizations of a random variable uniformly distributed over the unit cube $$P = (p_1, p_2, \cdots, p_n), p_i \in [0,1]^d$$. And we obtain the prefix sum $$S = (p_1, p_1+p_2, \cdots, \sum_{i=1}^n p_i)$$.

What is the probability that the convex hull of $$S$$ has all of the $$n$$ points of $$S$$ as extreme points?

I tried some brutal force enumeration with small n. Since $$\{S\}$$ is bounded by a Zonotope $$Z = \{G*P: G \in [0, 1]^n\}$$, i also plotted the $$Z$$

$$n=6, d=2, p_i \sim (Uniform(0, 1) \times Uniform(0, 1))$$, and on the left, it's density distribution of extreme points from all $$n!$$ different permutation and on the right, it's density distribution of extreme points of $$Z$$