\documentclass[12pt,titlepage]{article}

\usepackage{amsmath}
\usepackage{mathrsfs}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{mathtools}
\usepackage{graphicx}
\usepackage{color}
\usepackage{ucs}
\usepackage[utf8x]{inputenc}
\usepackage{xparse}
\usepackage{hyperref}

%----Macros----------
%
% Unresolved issues:
%
%  \righttoleftarrow
%  \lefttorightarrow
%
%  \color{} with HTML colorspec
%  \bgcolor
%  \array with options (without options, it's equivalent to the matrix environment)

% Of the standard HTML named colors, white, black, red, green, blue and yellow
% are predefined in the color package. Here are the rest.
\definecolor{aqua}{rgb}{0, 1.0, 1.0}
\definecolor{fuschia}{rgb}{1.0, 0, 1.0}
\definecolor{gray}{rgb}{0.502, 0.502, 0.502}
\definecolor{lime}{rgb}{0, 1.0, 0}
\definecolor{maroon}{rgb}{0.502, 0, 0}
\definecolor{navy}{rgb}{0, 0, 0.502}
\definecolor{olive}{rgb}{0.502, 0.502, 0}
\definecolor{purple}{rgb}{0.502, 0, 0.502}
\definecolor{silver}{rgb}{0.753, 0.753, 0.753}
\definecolor{teal}{rgb}{0, 0.502, 0.502}

% Because of conflicts, \space and \mathop are converted to
% \itexspace and \operatorname during preprocessing.

% itex: \space{ht}{dp}{wd}
%
% Height and baseline depth measurements are in units of tenths of an ex while
% the width is measured in tenths of an em.
\makeatletter
\newdimen\itex@wd%
\newdimen\itex@dp%
\newdimen\itex@thd%
\def\itexspace#1#2#3{\itex@wd=#3em%
\itex@wd=0.1\itex@wd%
\itex@dp=#2ex%
\itex@dp=0.1\itex@dp%
\itex@thd=#1ex%
\itex@thd=0.1\itex@thd%
\advance\itex@thd\the\itex@dp%
\makebox[\the\itex@wd]{\rule[-\the\itex@dp]{0cm}{\the\itex@thd}}}
\makeatother

% \tensor and \multiscript
\makeatletter
\newif\if@sup
\newtoks\@sups
\def\append@sup#1{\edef\act{\noexpand\@sups={\the\@sups #1}}\act}%
\def\reset@sup{\@supfalse\@sups={}}%
\def\mk@scripts#1#2{\if #2/ \if@sup ^{\the\@sups}\fi \else%
  \ifx #1_ \if@sup ^{\the\@sups}\reset@sup \fi {}_{#2}%
  \else \append@sup#2 \@suptrue \fi%
  \expandafter\mk@scripts\fi}
\def\tensor#1#2{\reset@sup#1\mk@scripts#2_/}
\def\multiscripts#1#2#3{\reset@sup{}\mk@scripts#1_/#2%
  \reset@sup\mk@scripts#3_/}
\makeatother

% \slash
\makeatletter
\newbox\slashbox \setbox\slashbox=\hbox{$/$}
\def\itex@pslash#1{\setbox\@tempboxa=\hbox{$#1$}
  \@tempdima=0.5\wd\slashbox \advance\@tempdima 0.5\wd\@tempboxa
  \copy\slashbox \kern-\@tempdima \box\@tempboxa}
\def\slash{\protect\itex@pslash}
\makeatother

% math-mode versions of \rlap, etc
% from Alexander Perlis, "A complement to \smash, \llap, and lap"
%   http://math.arizona.edu/~aprl/publications/mathclap/
\def\clap#1{\hbox to 0pt{\hss#1\hss}}
\def\mathllap{\mathpalette\mathllapinternal}
\def\mathrlap{\mathpalette\mathrlapinternal}
\def\mathclap{\mathpalette\mathclapinternal}
\def\mathllapinternal#1#2{\llap{$\mathsurround=0pt#1{#2}$}}
\def\mathrlapinternal#1#2{\rlap{$\mathsurround=0pt#1{#2}$}}
\def\mathclapinternal#1#2{\clap{$\mathsurround=0pt#1{#2}$}}

% Renames \sqrt as \oldsqrt and redefine root to result in \sqrt[#1]{#2}
\let\oldroot\root
\def\root#1#2{\oldroot #1 \of{#2}}
\renewcommand{\sqrt}[2][]{\oldroot #1 \of{#2}}

% Manually declare the txfonts symbolsC font
\DeclareSymbolFont{symbolsC}{U}{txsyc}{m}{n}
\SetSymbolFont{symbolsC}{bold}{U}{txsyc}{bx}{n}
\DeclareFontSubstitution{U}{txsyc}{m}{n}

% Manually declare the stmaryrd font
\DeclareSymbolFont{stmry}{U}{stmry}{m}{n}
\SetSymbolFont{stmry}{bold}{U}{stmry}{b}{n}

% Manually declare the MnSymbolE font
\DeclareFontFamily{OMX}{MnSymbolE}{}
\DeclareSymbolFont{mnomx}{OMX}{MnSymbolE}{m}{n}
\SetSymbolFont{mnomx}{bold}{OMX}{MnSymbolE}{b}{n}
\DeclareFontShape{OMX}{MnSymbolE}{m}{n}{
    <-6>  MnSymbolE5
   <6-7>  MnSymbolE6
   <7-8>  MnSymbolE7
   <8-9>  MnSymbolE8
   <9-10> MnSymbolE9
  <10-12> MnSymbolE10
  <12->   MnSymbolE12}{}

% Declare specific arrows from txfonts without loading the full package
\makeatletter
\def\re@DeclareMathSymbol#1#2#3#4{%
    \let#1=\undefined
    \DeclareMathSymbol{#1}{#2}{#3}{#4}}
\re@DeclareMathSymbol{\neArrow}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\neArr}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\seArrow}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\seArr}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\nwArrow}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\nwArr}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\swArrow}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\swArr}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\nequiv}{\mathrel}{symbolsC}{46}
\re@DeclareMathSymbol{\Perp}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\Vbar}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\sslash}{\mathrel}{stmry}{12}
\re@DeclareMathSymbol{\bigsqcap}{\mathop}{stmry}{"64}
\re@DeclareMathSymbol{\biginterleave}{\mathop}{stmry}{"6}
\re@DeclareMathSymbol{\invamp}{\mathrel}{symbolsC}{77}
\re@DeclareMathSymbol{\parr}{\mathrel}{symbolsC}{77}
\makeatother

% \llangle, \rrangle, \lmoustache and \rmoustache from MnSymbolE
\makeatletter
\def\Decl@Mn@Delim#1#2#3#4{%
  \if\relax\noexpand#1%
    \let#1\undefined
  \fi
  \DeclareMathDelimiter{#1}{#2}{#3}{#4}{#3}{#4}}
\def\Decl@Mn@Open#1#2#3{\Decl@Mn@Delim{#1}{\mathopen}{#2}{#3}}
\def\Decl@Mn@Close#1#2#3{\Decl@Mn@Delim{#1}{\mathclose}{#2}{#3}}
\Decl@Mn@Open{\llangle}{mnomx}{'164}
\Decl@Mn@Close{\rrangle}{mnomx}{'171}
\Decl@Mn@Open{\lmoustache}{mnomx}{'245}
\Decl@Mn@Close{\rmoustache}{mnomx}{'244}
\makeatother

% Widecheck
\makeatletter
\DeclareRobustCommand\widecheck[1]{{\mathpalette\@widecheck{#1}}}
\def\@widecheck#1#2{%
    \setbox\z@\hbox{\m@th$#1#2$}%
    \setbox\tw@\hbox{\m@th$#1%
       \widehat{%
          \vrule\@width\z@\@height\ht\z@
          \vrule\@height\z@\@width\wd\z@}$}%
    \dp\tw@-\ht\z@
    \@tempdima\ht\z@ \advance\@tempdima2\ht\tw@ \divide\@tempdima\thr@@
    \setbox\tw@\hbox{%
       \raise\@tempdima\hbox{\scalebox{1}[-1]{\lower\@tempdima\box
\tw@}}}%
    {\ooalign{\box\tw@ \cr \box\z@}}}
\makeatother

% \mathraisebox{voffset}[height][depth]{something}
\makeatletter
\NewDocumentCommand\mathraisebox{moom}{%
\IfNoValueTF{#2}{\def\@temp##1##2{\raisebox{#1}{$\m@th##1##2$}}}{%
\IfNoValueTF{#3}{\def\@temp##1##2{\raisebox{#1}[#2]{$\m@th##1##2$}}%
}{\def\@temp##1##2{\raisebox{#1}[#2][#3]{$\m@th##1##2$}}}}%
\mathpalette\@temp{#4}}
\makeatletter

% udots (taken from yhmath)
\makeatletter
\def\udots{\mathinner{\mkern2mu\raise\p@\hbox{.}
\mkern2mu\raise4\p@\hbox{.}\mkern1mu
\raise7\p@\vbox{\kern7\p@\hbox{.}}\mkern1mu}}
\makeatother

%% Fix array
\newcommand{\itexarray}[1]{\begin{matrix}#1\end{matrix}}
%% \itexnum is a noop
\newcommand{\itexnum}[1]{#1}

%% Renaming existing commands
\newcommand{\underoverset}[3]{\underset{#1}{\overset{#2}{#3}}}
\newcommand{\widevec}{\overrightarrow}
\newcommand{\darr}{\downarrow}
\newcommand{\nearr}{\nearrow}
\newcommand{\nwarr}{\nwarrow}
\newcommand{\searr}{\searrow}
\newcommand{\swarr}{\swarrow}
\newcommand{\curvearrowbotright}{\curvearrowright}
\newcommand{\uparr}{\uparrow}
\newcommand{\downuparrow}{\updownarrow}
\newcommand{\duparr}{\updownarrow}
\newcommand{\updarr}{\updownarrow}
\newcommand{\gt}{>}
\newcommand{\lt}{<}
\newcommand{\map}{\mapsto}
\newcommand{\embedsin}{\hookrightarrow}
\newcommand{\Alpha}{A}
\newcommand{\Beta}{B}
\newcommand{\Zeta}{Z}
\newcommand{\Eta}{H}
\newcommand{\Iota}{I}
\newcommand{\Kappa}{K}
\newcommand{\Mu}{M}
\newcommand{\Nu}{N}
\newcommand{\Rho}{P}
\newcommand{\Tau}{T}
\newcommand{\Upsi}{\Upsilon}
\newcommand{\omicron}{o}
\newcommand{\lang}{\langle}
\newcommand{\rang}{\rangle}
\newcommand{\Union}{\bigcup}
\newcommand{\Intersection}{\bigcap}
\newcommand{\Oplus}{\bigoplus}
\newcommand{\Otimes}{\bigotimes}
\newcommand{\Wedge}{\bigwedge}
\newcommand{\Vee}{\bigvee}
\newcommand{\coproduct}{\coprod}
\newcommand{\product}{\prod}
\newcommand{\closure}{\overline}
\newcommand{\integral}{\int}
\newcommand{\doubleintegral}{\iint}
\newcommand{\tripleintegral}{\iiint}
\newcommand{\quadrupleintegral}{\iiiint}
\newcommand{\conint}{\oint}
\newcommand{\contourintegral}{\oint}
\newcommand{\infinity}{\infty}
\newcommand{\bottom}{\bot}
\newcommand{\minusb}{\boxminus}
\newcommand{\plusb}{\boxplus}
\newcommand{\timesb}{\boxtimes}
\newcommand{\intersection}{\cap}
\newcommand{\union}{\cup}
\newcommand{\Del}{\nabla}
\newcommand{\odash}{\circleddash}
\newcommand{\negspace}{\!}
\newcommand{\widebar}{\overline}
\newcommand{\textsize}{\normalsize}
\renewcommand{\scriptsize}{\scriptstyle}
\newcommand{\scriptscriptsize}{\scriptscriptstyle}
\newcommand{\mathfr}{\mathfrak}
\newcommand{\statusline}[2]{#2}
\newcommand{\tooltip}[2]{#2}
\newcommand{\toggle}[2]{#2}

% Theorem Environments
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{prop}{Proposition}
\newtheorem{cor}{Corollary}
\newtheorem*{utheorem}{Theorem}
\newtheorem*{ulemma}{Lemma}
\newtheorem*{uprop}{Proposition}
\newtheorem*{ucor}{Corollary}
\theoremstyle{definition}
\newtheorem{defn}{Definition}
\newtheorem{example}{Example}
\newtheorem*{udefn}{Definition}
\newtheorem*{uexample}{Example}
\theoremstyle{remark}
\newtheorem{remark}{Remark}
\newtheorem{note}{Note}
\newtheorem*{uremark}{Remark}
\newtheorem*{unote}{Note}

%-------------------------------------------------------------------

\begin{document}

%-------------------------------------------------------------------


\section*{convex function}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{analysis}{}\paragraph*{{Analysis}}\label{analysis}

[[!include analysis - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

A \emph{convex function} is a [[real numbers|real]]-valued [[function]] defined on a [[convex set]] whose [[graph]] is the boundary of a [[convex set]].

There is another context where people say a function is convex if it is a [[Lipschitz function]] between [[metric spaces]] with Lipschitz constant (or Lipschitz modulus) 1. These are different concepts of convexity, although there are relations between convexity and Lipschitz continuity, as we shall see below.

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

Let $D$ be a [[convex space]]. A [[function]] $f \colon D \to \mathbb{R}$ is \textbf{convex} if the [[set]] $\{(x, y) \in D \times \mathbb{R}: y \geq f(x)\}$ is a convex subspace of $D \times \mathbb{R}$. Equivalently, $f$ is \textbf{convex} if for all $x, y \in D$,

\begin{displaymath}
f(t x + (1-t) y) \leq t f(x) + (1-t) f(y)
\end{displaymath}
whenever $0 \leq t \leq 1$.

This definition obviously extends to functions $f \colon D \to I$ where $I$ is an [[interval]] of $\mathbb{R}$ (whether open or closed or half-open, it doesn't matter).

The function $f$ is called \textbf{concave} if it satisfy the reverse inequality to the one given above.

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{itemize}%
\item A [[homomorphism]] of [[convex sets]], i.e. a convex-linear map, $D \to \mathbb{R}$, is of course convex.


\item The [[norm]] function $\mathbb{C} \to \mathbb{R}$ is convex. This follows readily from multiplicativity ${|x y|} = {|x|} \cdot {|y|}$ and the [[triangle inequality]] ${|x + y|} \leq {|x|} + {|y|}$.


\item More generally, for a [[normed vector space]] $V$, the norm function ${\|(-)\|}$ is convex, again by the triangle inequality and the scaling axiom ${\|\alpha v\|} = {|\alpha|} \cdot {\|v\|}$ for [[scalars]] $\alpha$.


\item For any twice-[[differentiable function]] $f \colon (a, b) \to \mathbb{R}$, the second derivative $f''$ is nonnegative iff $f$ is convex; this may be proven using the [[mean value theorem]]. Examples include the [[exponential function]] $\exp: (-\infty, \infty) \to \mathbb{R}$ and the $p$-power function $[0, \infty) \to \mathbb{R}: t \mapsto t^p$ if $p \geq 1$.


\item More generally, for an open convex region $D \subseteq \mathbb{R}^n$ in a [[Euclidean space]], a twice-differentiable function $f: D \to \mathbb{R}$ is convex iff its [[Hessian]] is a [[semidefinite element|positive semidefinite]] [[bilinear form]].


\item Any positive $\mathbb{R}$-[[linear combination]] of convex functions on $D$ is again convex.


\item The pointwise [[maximum]] of two convex functions on $D$ is again convex.


\item If $g: V \to W$ is a homomorphism or convex-linear map between [[convex spaces]], and if $f: W \to \mathbb{R}$ is convex, then $f \circ g: V \to \mathbb{R}$ is convex.



\end{itemize}
In the next two examples, $I \subseteq \mathbb{R}$ is an [[interval]].

\begin{itemize}%
\item It is not generally true that a composition $D \stackrel{g}{\to} I \stackrel{f}{\to} \mathbb{R}$ of convex functions is convex. For example, this fails for the case $D = \mathbb{R}$ and $g(x) = x^2 + 1$ and $f: [1, \infty) \to \mathbb{R}$ given by $f(x) = x^{-1}$.


\item However, if $f: I \to \mathbb{R}$ is both [[monotone function|monotone increasing]] and convex, then for any convex function $g: D \to I$, the composite $f \circ g: D \to \mathbb{R}$ is convex (as is easily shown).



\end{itemize}
A special case of the last class of examples are functions of the form $e^f = \exp \circ f$ where $f$ is convex. We say a function $f: D \to (0, \infty)$ is \emph{log-convex} if $\log f$ is convex. We see then that log-convex functions are also convex.

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\begin{lemma}
\label{squeeze}\hypertarget{squeeze}{}
If $f: (a, b) \to \mathbb{R}$ is convex and $u \lt v \lt w \in (a, b)$, then

\begin{displaymath}
\frac{f(u) - f(v)}{u - v} \leq \frac{f(u) - f(w)}{u - w} \leq \frac{f(v) - f(w)}{v - w}.
\end{displaymath}
Consequently, the function $x \mapsto \frac{f(x) - f(v)}{x - v}$ on the domain $u \lt x \lt v$ is increasing and is bounded above by $\frac{f(v)-f(w)}{v-w}$; similarly, this function on the domain $v \lt x \lt w$ is increasing and bounded below by $\frac{f(u)-f(v)}{u-v}$.

\end{lemma}
The proof is virtually trivial; just write $v$ as a convex combination $t u + (1-t)w$ and use the definition of convexity.

\begin{prop}
\label{}\hypertarget{}{}
A convex function $f: (a, b) \to \mathbb{R}$ is continuous.

\end{prop}
\begin{proof}
For any point $x_0 \in (a, b)$ there are $s, t, u, v \in (a, b)$ with $s \lt t \lt x_0 \lt u \lt v$. Then for any $x \in (t, x_0) \cup (x_0, u)$ we have by Lemma \ref{squeeze}

\begin{displaymath}
\frac{f(s)-f(t)}{s-t} \leq \frac{f(x) - f(x_0)}{x-x_0} \leq \frac{f(u)-f(v)}{u-v}
\end{displaymath}
so that $K = \max\{\left\vert \frac{f(s)-f(t)}{s-t}\right\vert, \left\vert \frac{f(u)-f(v)}{u-v}\right\vert\}$ serves as a Lipschitz constant, i.e., we have

\begin{displaymath}
|f(x) - f(x_0)| \leq K|x - x_0|
\end{displaymath}
for all $x \in (t, u)$. This is enough to force continuity at the point $x_0$.

\end{proof}
In the converse direction, we have the following result which frequently arises in practice.

\begin{prop}
\label{}\hypertarget{}{}
If $D \subseteq \mathbb{R}^n$ is a convex set and $f: D \to \mathbb{R}$ is a continuous function such that

\begin{displaymath}
f\left(\frac{x+y}{2}\right) \leq \frac{f(x) + f(y)}{2},
\end{displaymath}
then $f$ is convex.

\end{prop}
\begin{proof}
For any fixed $x, y \in D$, the function $h: [0, 1] \to \mathbb{R}$ defined by

\begin{displaymath}
h(t) = t f(x) + (1-t)f(y) - f(t x + (1-t)y)
\end{displaymath}
is continuous. The [[inverse image]] $h^{-1}([0, \infty))$ is closed in $[0, 1]$ and, by an easy induction argument, contains the set of [[dyadic rational numbers]] $t \in [0, 1]$, which is dense in $[0, 1]$. Being closed and dense, $h^{-1}([0, \infty))$ is all of $[0, 1]$, i.e., $h(t) \geq 0$ for all $t \in [0, 1]$, but this is precisely the condition that $f$ is convex.

\end{proof}
Another easy consequence of Lemma \ref{squeeze} is

\begin{prop}
\label{}\hypertarget{}{}
For a convex function $f: (a, b) \to \mathbb{R}$, the right-hand and left-hand derivatives

\begin{displaymath}
\underset{x \searrow x_0}{\lim} \frac{f(x)-f(x_0)}{x-x_0}, \qquad \underset{x \nearrow x_0}{\lim} \frac{f(x)-f(x_0)}{x-x_0}
\end{displaymath}
of $f$ exist at every point $x_0 \in (a, b)$, and the left-hand derivative at $x_0$ is less than or equal to the right-hand derivative at $x_0$.

\end{prop}
It further follows from Lemma \ref{squeeze} that if $f$ is convex and we define $(D f)(x_0)$ to be the average of the right-hand and left-hand derivatives, then $D f$ is monotone increasing and hence is discontinuous at at most countable many points of jump discontinuity (whence $D f$ is [[Riemann integration|Riemann-integrable]], for instance).

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[convex analysis]]


\item [[Legendre transform]]


\item [[Lipschitz function]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

See also

\begin{itemize}%
\item Wikipedia, \emph{\href{http://en.wikipedia.org/wiki/Convex_function}{Convex function}}

\end{itemize}
[[!redirects convex map]] [[!redirects convex maps]] [[!redirects convex function]] [[!redirects convex functions]] [[!redirects log-convex function]] [[!redirects log-convex functions]]

[[!redirects concave map]] [[!redirects concave maps]] [[!redirects concave function]] [[!redirects concave functions]] [[!redirects log-concave function]] [[!redirects log-concave functions]]



\end{document}