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?
Best Answer
They're affinely independent if none of them is in the affine space spanned by the others, i.e. the smallest affine space containing the others. Three points are in a plane, so the fourth point must not be in the same plane as the first three.