Affine combinations




     What  properties do lines have in 2‐space and planes have in
3‐space that would be useful in higher dimensions?

The hyperplanes will be important for understanding the  multidi‐
mensional nature of the linear programming problems.

What  would  be  the analogue of a polyhedron "look like" in more
than three dimensions?

A partial answer is provided by  two‐dimensional  projections  of
the four dimensional object, created in a manner analogous to two
dimensional projections of a three‐dimensional object.

Given vectors v1,v2,v3, ...



affine hull



convex combination and convex hull:



The affine hull of distinct points v1 and v2 is the line



y‐v1 is a linear combination:




The convex hull is the line segment.



Affine set:
















                               ‐2‐


affine‐invariant property


     Self‐concordance  is  an affine‐invariant property, i.e., if
we apply a linear transformation of variables to  a  self‐concor‐
dant  function,  we obtain a self‐concordant function. (Therefore
the complexity estimate that we obtain for  Newton’s  method  ap‐
plied  to  a  self‐concordant  function  is independent of affine
changes of coordinates.)

What about convexity? If we  apply  a  linear  transformation  of
variables to a convex function, do we obtain a convex function?



operations that preserve convexity




     1. intersection

     2. affine functions