\documentclass[12pt]{article}

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\usepackage{fullpage,amsmath,amssymb,amsthm,framed,graphicx,color}
\newtheorem{theorem}{Theorem}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
\usepackage{bm}
\usepackage{pifont}% http://ctan.org/pkg/pifont
\newcommand{\cmark}{\ding{51}}%
\newcommand{\xmark}{\ding{55}}%

\def\NN{\mathbb{N}}
\def\ZZ{\mathbb{Z}}
\def\RR{\mathbb{R}}
\def\QQ{\mathbb{Q}}
\def\CC{\mathbb{C}}
\def\KK{\mathbb{K}}


\def\cA{\mathcal{A}}
\def\cB{\mathcal{B}}
\def\cC{\mathcal{C}}
\def\cF{\mathcal{F}}
\def\cH{\mathcal{H}}
\def\cL{\mathcal{L}}
\def\cN{\mathcal{N}}
\def\cO{\mathcal{O}}
\def\cP{\mathcal{P}}
\def\cR{\mathcal{R}}
\def\cS{\mathcal{S}}

\def\ee{\varepsilon}
\def\ii{\mathfrak{i}}

\newcommand{\mb}[1]{\ensuremath{\mathbf{ #1 }}}
\newcommand{\norm}[1]{\ensuremath{\left\| #1\right\| }}
\newcommand{\enorm}[1]{\ensuremath{|\!|\!| #1|\!|\!| }}

\DeclareMathOperator*{\esssup}{ess\,sup}
\DeclareMathOperator*{\supp}{supp}
\DeclareMathOperator*{\Ker}{Ker}
\DeclareMathOperator*{\Range}{Im}
\DeclareMathOperator*{\spn}{span}

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\begin{document}
\noindent
\rule{\textwidth}{0.05cm}
\begin{center}
{{\Large\bf Section 2.20:} \bf A, 1, 6}
\end{center}
\rule[0.2cm]{\textwidth}{0.05cm}

\begin{enumerate}


\item[A.]  Let $A$ be a convex set in a real vector space.  Let
  $f:A\to\RR$ be a convex function, and let $g: f(A)\to\RR$ be a
  convex and increasing function.  Show that $h=g\circ f: A\to\RR$ is
  a convex function.
  
\item[1.] Let $(X,\|\cdot\|)$ be a normed vector space, and let $p\geq
  1$.  Show that $h:X\to\RR$ defined by $h(x)=\|x\|^p$ is a convex
  function.


\item[6.]  Show that a strictly convex finite-dimensional normed
  vector space is uniformly convex.
   \textbf{Hint:} You may wish to argue that
\begin{align*}
  K(\ee)\doteq\{(x,y)\in X\times X:\,\|x\|=\|y\|=1 \mbox{ and }
  \|x-y\|\geq \ee\}
\end{align*}
is compact in $X\times X$ (under the standard product topology).

\end{enumerate}


\end{document}