Ref:

• “Linear Algebra” by Lax.

ROUGH NOTES (!)
Updated: 15/3/26

We will study

\[{ \boxed{\textbf{Convexity}} }\] \[{ }\] \[{ \underline{\textbf{Convex Sets}} }\]

Def [Line Segment]
Let ${ X }$ be a (real) vector space. Let ${ x, y \in X . }$
The line segment with endpoints ${ x }$ and ${ y }$ is

\[{ [x, y] = \lbrace x + a ( y - x ) : 0 \leq a \leq 1 \rbrace . }\]

Def [Convex Set]
Let ${ X }$ be a (real) vector space. Let ${ K \subseteq X . }$
We say ${ K }$ is convex if whenever ${ x, y \in K }$ we have ${ [x, y] \subseteq K . }$

We can freely take line segments in a convex set. Note that intuitively convex sets must be “round”.

Eg [Convex Sets]
Let ${ X }$ be a vector space. Then ${ X, }$ ${ \emptyset , }$ ${ \lbrace x \rbrace }$ for ${ x \in X , }$ any line segment are all convex sets.

Let ${ X }$ be a vector space. Let ${ \ell : X \to \mathbb{R} }$ be a linear function. Then the sets

\[{ {\begin{aligned} &\, \lbrace \ell(x) = c \rbrace \quad (\text{a hyperplane}), \\ &\, \lbrace \ell(x) < c \rbrace \quad (\text{an open half-space}), \\ &\, \lbrace \ell(x) \leq c \rbrace \quad (\text{a closed half-space}) \end{aligned}} }\]

are all convex sets.

Let ${ X }$ be the space of all polynomials (with real coefficients). Let ${ K }$ be the subset of all polynomials that are positive at every point of ${ (0,1) . }$ Note that ${ K }$ is convex.

Suppose polynomials ${ p(t) > 0 }$ and ${ q(t) > 0 }$ for all ${ t \in (0, 1) . }$ Let ${ \alpha \in [0, 1] . }$ Note that ${ (\alpha p + (1 - \alpha) q) (t) > 0 }$ for all ${ t \in (0, 1) . }$

Let ${ X }$ be the space of real symmetric matrices. Let ${ K }$ be the subset of positive matrices. Note that ${ K }$ is convex.

Suppose symmetric matrices ${ A , B > 0 . }$ Let ${ \alpha \in [0, 1] . }$ Note that ${ (\alpha A + (1 - \alpha) B) > 0 . }$

Obs [Intersection of Convex Sets]

The intersection of any collection of convex sets is convex.

Pf: Let ${ X }$ be a vector space. Let ${ \lbrace C _{\alpha} : \alpha \in A \rbrace }$ be a collection of convex sets.

Let ${ x, y \in \cap C _{\alpha} . }$ Hence ${ x, y \in C _{\alpha} }$ for all ${ \alpha \in A . }$ Hence ${ [x, y] \subseteq C _{\alpha} }$ for all ${ \alpha \in A . }$ Hence ${ [x, y] \subseteq \cap C _{\alpha} . }$

Hence ${ \cap C _{\alpha} }$ is convex. ${ \blacksquare }$

Obs [Sum of Convex Sets]

The sum of two convex sets is convex.

Pf: Let ${ X }$ be a vector space. Let ${ H, K }$ be convex sets. Is ${ H + K }$ convex?

Let ${ h _1 + k _1 , h _2 + k _2 \in H + K . }$ Let ${ \alpha \in [0, 1] . }$ Note that ${ \alpha (h _1 + k _1) + (1 - \alpha) (h _2 + k _2) }$ ${ = (\alpha h _1 + (1 - \alpha) h _2) + (\alpha k _1 + (1 - \alpha) k _2) }$ is also in ${ H + K . }$

Hence ${ H + K }$ is convex. ${ \blacksquare }$

Def [(Algebraic) Interior Point]

Let ${ X }$ be a vector space. Let ${ S \subseteq X . }$

A point ${ x }$ is called an interior point of ${ S }$ if for every ${ y \in X , }$ ${x + yt \in S }$ for all sufficiently small positive ${ t . }$

Let ${ x \in S }$ be an interior point. Intuitively, we can perturb ${ x }$ a little bit along any line, and still stay within ${ S . }$

Def [(Algebraic) Open Convex Set]

Let ${ X }$ be a vector space. Let ${ K }$ be a convex set.

A convex set ${ K }$ is open if every point in it is an interior point.