Can convex hulls contain duplicate points? Given 4 vertices:
(-3.2, 0.8), (-3.2, -0.8), (3.2, -0.8), (3.2, 0.8)
A function which I did not write, and given vertices, will return the points of a convex hull.
Using the points above, it returns these points:
(-3.2, 0.8), (-3.2, -0.8), (3.2, -0.8), (3.2, 0.8), (-3.2, 0.8)
Is it improper to have a duplicate point? 
Sometime tells me it is, but I'd hate to suggest to the author that he made a mistake when he did not.
 A: The convex hull (call this $H$) of points $P_1,\cdots P_n$ is the set of combinations
$$\sum_{k=1}^n c_kP_k$$
where each $c_k \ge 0$ and the sum of the $c_k$ is 1. If a copy of one of the points, say $P_n$ is adjoined to the list $P_1,\cdots P_n$, the now possibly different hull, call it $H'$ is by the above definition the set of combinations
$$c_{n+1}P_n+\sum_{k=1}^n c_kP_k$$
where now all the $c_k,\ k=1,2,...,n+1$ are nonnegative and their sum is $1$.
Now the claim is that in fact $H'=H$ as sets, i.e. the hulls are the same. It's clear that $H \subset H'$ by taking $c_{n+1}=0$. 
For the reverse direction, define a new coefficient name $e_n=c_n+c_{n+1}$ and note that now $H'$ is given by the set of combinations
$$\sum_{k=1}^{n-1} c_kP_k+e_nP_n.$$
The constraints on these $n$ coefficients are again that they be each nonnegative and sum to $1$. This then shows that $H' \subset H$, and we arrive at the claimed statement, that adding a copy of a point in describing a convex hull has no effect on the hull generated.
