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\def\rit{ \hbox{\it I\hskip -2pt R} }

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\noindent{\bf Math 273b}: Calculus of Variations 

\noindent{\bf Homework \#1}

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\noindent{\bf [1]} Let $V$ be a real vector space and $C$ a convex nonempty subset of $V$. Suppose that $f:C\mapsto \rit$ is a convex function. Show that the set of global minimizers of $f$ is a convex set. 

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\noindent{\bf [2]} Let $V$ be a real vector space and $F:V\rightarrow \overline{\rit}$ be a convex function, thus for every $u,v\in V$, we have 
$$F(\lambda u+(1-\lambda)v)\leq \lambda F(u)+(1-\lambda)F(v),$$ 
$\forall\lambda\in [0,1]$, whenever the RHS is defined (the RHS is not defined when $F(u)=-F(v)=+\infty$ or $F(u)=-F(v)=-\infty$). 

(a) If $F$ is convex, show that for every $u_1,...,u_n$ of points of $V$ and for every family $\lambda_1,...,\lambda_n$, $\lambda_i\geq0$, $\sum_{i=1}^n\lambda_i=1$, we have 
$$F(\sum_{i=1}^n\lambda_i u_i)\leq\sum_{i=1}^n\lambda_iF(u_i).$$ 

(b) If $F:V\rightarrow \overline{\rit}$ is convex, show that the sections 
$\{u: \ F(u)\leq a\}$ and $\{u: \ F(u)<a\}$ are convex sets. 
Show that the converse is not true. 

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\noindent{\bf [3]} The {\it epigraph} of a function $F:V\rightarrow \rit$ is the 
set 
$$\mbox{epi}F=\{(u,a)\in V\times \rit|\ f(u)\leq a\},$$
where $V$ is a normed vector space.
Show that the function $F$ is convex if and only if its epigraph is convex.

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\noindent{\bf [4]} Note that a function $F:V\mapsto \overline{\rit}$, with $V$ a normed vector space, is lower semi-continuous (l.s.c.) on $V$ by the equivalent definition:
$$\forall a\in \rit: \ \ \Big\{u\in V| \ F(u)\leq a\Big\} \mbox{ is closed.}$$

Using this (or another method), show that $F$ is l.s.c. iff its epigraph is closed (hint: consider
 the function on $V\times \rit$ defined by $f(u,a)=F(u)-a$).

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\noindent{\bf [5]} Let $F:\rit^n\rightarrow\rit$ be continuous and coercive with respect to the Euclidean norm $\|x\|$, for $x\in\rit^n$. Show: then there is $x_0\in\rit^n$ such that $F(x_0)=\inf_{x\in \rit^n} F(x)$. If in addition $F$ is strictly convex, then the minimizer $x_0$ is unique (give a simplified version of the proof given in class for the more general case). 

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