I got the following question in a research project.
Consider n vectors x1,...,xn in R^m satisfying the following property (*):
(*) no vector is coordinate-wise smaller than any convex combination of the other n-1 vectors.
I want to show that for any epsilon>0, I can find another n vectors y1,...,yn such that
(1) y1 is in the epsilon-ball of x1, y2 is in the epsilon-ball of x2,......,yn is in the epsilon-ball of xn;
(2) y1,...,yn still satisfy (*)
(3) each vector of y1,...,yn is outside the convex hull of the other (n-1) vectors.
I believe it's true and I also see how it works when m=2.
I greatly appreciate your help.