convex set

constituents

A set $C$ is a convex set if:

\begin{equation} x_1, x_2 \in C, 0 \leq \theta \leq 1 \implies \theta x_{1} + \qty(1-\theta) x_{2} \in C \end{equation}

All points of the form $x = \theta x_{1} + \qty(1-\theta) x_{2}$, with $\theta \in \mathbb{R}$ is a “line through $x_1$, $x_2$”.

For set $G$, for all two points $x_1, x_2 \in G$, all points lying on the line $x_1, x_2 \in G$. That is , $$

For instance, the solution set of a set of linear equations $\qty {x \mid A x = b}$.

all points form $x = \theta x_{1} + \qty(1-\theta)x_{2}$, with $0 \leq \theta \leq 1$.

A convex combination of points $x_1 … x_{k}$ has

\begin{equation} x = \theta_{1} x_1 + \theta_{2} x_{2} + \dots \theta_{k} x_{k} \end{equation}

with $\sum_{j}^{} \theta_{j} = 1$.

set of all convex combination of points in $S$

Conic combination any combination of the form

\begin{equation} x = \theta_{1} x_{1} + \theta_{2} x_{2} \end{equation}

with $\forall \theta_{j} > 0$.

convex cone is the set that contains all conic combinations of points in the set

Set of the form:

\begin{equation} \qty {x \mid a^{T} x = b}, a \neq 0 \end{equation}

\begin{equation} \qty {x \mid a^{T}x \leq b}, a \neq 0 \end{equation}