Lecture 3: Convex sets
Rajat Mittal ?
IIT Kanpur
We denote the set of real numbers as R. Most of the time we will be working with space Rnand its
elements will be called vectors. Remember that a subspace is a set of vectors closed under addition and
scalar multiplication.
First we learn how to take interesting combinations of a given set of vectors. For vectors x1,x2,· · · , xk;
any point yis a linear combination of them iff
y=α1x1+α2x2· · · +αkxk∀i, αi∈R
If we restrict αi’s to be positive then we get something called a conic combination.
y=α1x1+α2x2· · · +αkxk∀i, αi≥0∈R
Instead of being positive, if we put the restriction that αi’s sum up to 1, it is called an affine combination
y=α1x1+α2x2· · · +αkxk∀i, αi∈R,X
i
αi= 1
When a combination is affine as well as conic, it is called a convex combination.
y=α1x1+α2x2· · · +αkxk∀i, αi≥0∈R,X
i
αi= 1
Exercise 1. What is the linear/conic/affine/convex combination of two points in R2?
1 Affine sets
Lets start by defining an affine set.
Definition 1. A set is called “affine” iff for any two points in the set, the line through them is contained in
the set. In other words, for any two points in the set, their affine combinations are in the set itself.
Theorem 1. A set is affine iff any affine combination of points in the set is in the set itself.
Proof. Exercise. (Use induction)
Exercise 2. What is the affine combination of three points?
Suppose the three points are x1, x2, x3. Then any affine combination can be written as θ1x1+θ2x2+
θ3x3,Piθi= 1. We can write the expression as θ1(x1−x3)+ θ2(x2−x3)+x3. Since θ1, θ2are unconstrained
now, this sum can be thought of intuitively as (x3+ plane generated by x1−x3and x2−x3). This gives a
nice relation between affine and linear combinations.
1.1 Examples of affine sets
–Offset + Subspace: Any affine set can thought of as an offset xadded to some subspace S∼(V=
{v+x:∼v∈S}). It is easy to see that such a set is affine. Also given an affine set Vand a point inside
it x; it can be shown that S={v−x:v∈V}is a subspace. Notice that the subspace associated with
an affine set does not depend upon the point chosen (exercise). So we can define the dimension of the
affine set as the dimension of the subspace. So it turns out that offset+subspace is not just an example,
but a characterization of affine sets.
–Exercise 3. Show that the feasible region of a set of linear equations is affine.
?Thanks to books from Boyd and Vandenberghe, Dantzig and Thapa, Papadimitriou and Steiglitz
