In Stephen Boyd's book on Convex optimization he points out that k+1 affinely independent points form a simplex with affine dimension k.
My understanding of affinely independent points is that no 3 points are in a line. So if I take 4 points no 3 of which are in a line in $R^2$ than I get a simplex of affine dimension 3.
How is it possible for a set to have dimension more than 2 in $R^2$?
Please correct me if I am wrong.
On further inspection I realized that Boyd says "affine dimension of simplex". Now simplex is a convex set and affine dimension should be defined for an affine set. Isn't that correct?