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.