|
|
|
Viewing Version
9
of
'convex function'
|
[ view 'convex function'
|
back to history
]
| Title of object: |
convex function |
| Canonical Name: |
ConvexFunction |
| Type: |
Definition |
| Created on: |
2001-10-15 22:34:11 |
| Modified on: |
2003-10-18 04:47:55 |
| Classification: |
msc:26B25, msc:26A51 |
| Defines: |
convex function, concave function, strictly convex function, strictly concave function |
Preamble:
% this is the default PlanetMath preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.
% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
%\usepackage{amsthm}
% making logically defined graphics
%\usepackage{xypic}
% there are many more packages, add them here as you need them
% define commands here
\newcommand{\sR}[0]{\mathbb{R}}
\newcommand{\sC}[0]{\mathbb{C}}
\newcommand{\sN}[0]{\mathbb{N}}
\newcommand{\sZ}[0]{\mathbb{Z}}
% The below lines should work as the command
% \renewcommand{\bibname}{References}
% without creating havoc when rendering an entry in
% the page-image mode.
\makeatletter
\@ifundefined{bibname}{}{\renewcommand{\bibname}{References}}
\makeatother |
Content:
{\bf Definition} Suppose $\Omega$ is a convex set in a vector space over $\sR$
(or $\sC$), and suppose $f$ is a function $f:\Omega\to \sR$.
If for any $x,y\in \Omega$ and any $\lambda \in (0,1)$, we have
$$f\Big( \lambda a + (1-\lambda)b\Big)\leq \lambda f(a)+(1-\lambda)f(b),$$
we say that $f$ is a {\bf convex function}.
If for any $x,y\in \Omega$ and any $\lambda \in (0,1)$, we have
$$f\Big( \lambda a + (1-\lambda)b\Big)\geq \lambda f(a)+(1-\lambda)f(b),$$
we say that $f$ is a {\bf concave function}. If either of the inequalities
are strict, then we say that $f$ is a {\bf strictly convex function},
or a {\bf strictly concave function}, respectively.
\subsubsection{Properties}
\begin{itemize}
\item A function $f$ is a (strictly) convex function if and only if $-f$ is
a (strictly) concave function.
\item On $\sR$, a continuous function is convex
if and only if for all $x,y\in \sR$, we have
$$f\left(\frac{x+y}{2}\right)\le\frac{f(x)+f(y)}{2}.$$
\item A twice continuously differentiable function on $\sR$ is convex if
and only if $f''(x) \ge 0$ for all $x\in \sR$.
\end{itemize}
\subsubsection{Examples}
\begin{itemize}
\item $e^x$,$e^{-x}$, and $x^2$ are convex functions on $\sR$.
\item A norm is a convex function.
\item On $\sR^2$, the $1$-norm and the $\infty$-norm
(i.e., $||(x,y)||_1=|x|+|y|$ and $||(x,y)||_\infty=\max\{|x|,|y|\}$) are not strictly convex
(\cite{kreyszig}, pp. 334-335).
\end{itemize}
\begin{thebibliography}{9}
\bibitem{kreyszig} E. Kreyszig,
\emph{Introductory Functional Analysis With Applications},
John Wiley \& Sons, 1978.
\end{thebibliography} |
|
|
|
|
|
