Euclidian Geometry Crash Course — Jemoka Knowledge Base

line All points of the form x = \theta x_{1} + \left(1-\theta\right) x_{2}, with \theta \in \mathbb{R} is a “line through x_1, x_2”. affine set 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. For instance, the solution set of a set of linear equations \left\{x \mid A x = b\right\}. convex set convex set, line segment all points form x = \theta x_{1} + \left(1-\theta\right)x_{2}, with 0 \leq \theta \leq 1. convex combination 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, \forall \theta_{j} \geq 0. convex hull set of all convex combination of points in S

\begin{equation} \left\{\theta_{1} x_1 + \dots + \theta_{k}x_{k} \mid x_{i} \in C, \theta_{i} \geq 0, i = 1 \dots k, \sum_{j}^{}\theta_{j} = 1\right\} \end{equation}

convex cone Conic combination any combination of the form

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

with \forall \theta_{j} \geq 0. More generally a combination \theta_{1} x_1 + … + \theta_{n} x_{n} for \forall \theta_{j} \geq 0 is called a “conic combination”. convex cone or a conic hull is the set of all conic combinations of points in C:

\begin{equation} \left\{\theta_{1} x_1 + \dots + \theta_{k}x_{k} \mid x_{i} \in C, \theta_{i} \geq 0, i = 1 \dots k\right\} \end{equation}

properties of convex cone closed under non-negative scaling closed under addition hyperplane Set of the form:

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

“the solution set f a single linear equation”. If b were 0, we can think about it as “all the points that are orthogonal to a. affine set convex set halfspace \begin{equation} \left{x \mid a^{T}x \leq b\right}, a \neq 0 \end{equation} convex set Euclidian ball Set of points with L_2 norm smaller than r:

\begin{equation} B\left(x_{c}, r\right) = \left\{x \mid \norm{x -x_{c}}_{2} \leq r\right\} = \left\{x_{c} + ru \mid \norm{u}_{2} \leq 1\right\} \end{equation}

ellipsoid For Symmetric PSD P \in S_{++}^{n}:

\begin{equation} \left\{x \mid \left(x- x_{c}\right)^{T} P^{-1} \left(x - x_{c}\right) \leq 1\right\} \end{equation}

also written as:

\begin{equation} \left\{x_{c} + A u \mid \norm{u}_{2} \leq 1\right\} \end{equation}

for nonsigular A. norm properties of the norm norm ball \begin{equation} \left{x \mid \norm{x - x_{c}} \leq r\right} \end{equation} norm cone \begin{equation} \left{\left(x,t\right) \mid \norm{x} \leq t\right} \end{equation} polyhedron \begin{equation} \left{x \mid Ax \preceq b, Cx = d\right} \end{equation} where \preceq is “elementwise less-than”. You can think of each a_{j}^{T} as a row of A. PSD cones symmetric matrices positive semidefinite matrices (geometry) \begin{equation} S_{+}^{n} = \left{X \in S^{n} \mid X \succeq 0\right} \end{equation} this is a convex cone. positive definite (symmetric) metricies (geometry) \begin{equation} S^{n}_{++} = \left{X \in S^{n} \mid X \succ 0\right} \end{equation} proper cone a convex cone K \subseteq R^{n} is a proper cone if: K is closed K is solid (non-empty interior) K is pointed (contains no line) — v and -v can’t both have a line K is non-trivial (not empty) generalized inequality For proper cone K:

\begin{equation} x \preceq_{K} y \Leftrightarrow y - x \in K \end{equation}

\begin{equation} x \prec_{k} y \Leftrightarrow y - x \in \text{interior} K \end{equation}

Triangle inequality holds:

\begin{equation} x \preceq_{k} y, u \preceq_{k} v \implies x+u \preceq_{k} y + v \end{equation}

This is not well ordered.