\(\DeclarePairedDelimiterX{\Set}[2]{\{}{\}}{#1 \nonscript\;\delimsize\vert\nonscript\; #2}\) \( \DeclarePairedDelimiter{\set}{\{}{\}}\) \( \DeclarePairedDelimiter{\parens}{\left(}{\right)}\) \(\DeclarePairedDelimiterX{\innerproduct}[1]{\langle}{\rangle}{#1}\) \(\newcommand{\ip}[1]{\innerproduct{#1}}\) \(\newcommand{\bmat}[1]{\left[\hspace{2.0pt}\begin{matrix}#1\end{matrix}\hspace{2.0pt}\right]}\) \(\newcommand{\barray}[1]{\left[\hspace{2.0pt}\begin{matrix}#1\end{matrix}\hspace{2.0pt}\right]}\) \(\newcommand{\mat}[1]{\begin{matrix}#1\end{matrix}}\) \(\newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}}\) \(\newcommand{\mathword}[1]{\mathop{\textup{#1}}}\)
Real Halfspaces
Real Convex Sets
Real Polyhedra
Needed by:
Sheet PDF
Graph PDF

Real Convex Sets and Halfspaces

Main result

Let $(b_i)_{i \in I}$ be a family in $\R ^n$ and $(\beta _i)_{i \in I}$ be a family in $\R $. The set

\[ \Set*{x \in \R ^n}{\ip{x,b_i} \leq \beta _i \text{ for all } i \in I} \]

is convex.

A polyhedral convex set is one which can be expressed as the intersection of a finite family of closed halfspaces.

Copyright © 2023 The Bourbaki Authors — All rights reserved — Version 13a6779cc About Show the old page view