Extreme points of prefix sum and prefix sum of subsets.

Given an ordered set of points $P = (p_1, p_2, cdots, p_n)$, where $p_i in R^d$.

The prefix sum set $S$ of $P$ is defined as
$$
S = (p_1, p_1+p_2, p_1+p_2+p_3, cdots, sum_{i=1}^n p_i)
$$

And there exists a convex hull of $S$ denote $CH(S)$, a subset of elements of $S$ on the boundary of $CH(S)$ is referred to as extreme points $E = Boudardy(CH(S)) cap S$.

$E := f(P)$

The interest is that given an index set $I$ of $P$ and $E$, How can I find the extreme points of it namely $f(P_I)$ as efficient as possible?

E.g. $d=2, n=5$.

$$
P = ([0, 1],
[2, 1],
[0, 2],
[2, 3],
[2, 7]) \
S = ([0, 1],
[ 2, 2],
[ 2, 4],
[ 4, 7],
[ 6, 14]) \
E = (
[ 0, 1],
[ 2, 2],
[ 4, 7], [ 6, 14])\
$$
Given $I= (0, 1, 2)$

Then $P_I = ([0, 1],
[2, 1],
[0, 2],)$, $S_I=([0, 1],
[ 2, 2],
[ 2, 4],)$ and $E_I$ was simply be the same as $S_I$ .

Another example, in a $d=2$ case, if the ordering of $P$ is the same as the ordering of $tan^{-1}(p_i[1], p_i[0])$, then $E_I$ is always going to be the same as S_I.