Affine transformations and convexity

The purpose of this document is to prove some basic properties of affine transformations involving convex sets. Here are a few online references for background information: http://math.ucr.edu/∼res/progeom/pgnotes02.pdf http://math.ucr.edu/∼res/math133/metgeom.pdf Recall that an affine transformation of R is a map of the form F (x) = b+A(x), where b ∈ E is some fixed vector and A is an invertible linear tranformation of R. Affine transformations satisfy a weak analog of the basic identities which characterize linear transformations. LEMMA 1. Let F as above be an affine transformation, let x0, · · · ,xk ∈ R , and suppose that t0, · · · , tk ∈ R satisfy ∑