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\lhead{\tiny Symplectic geometry, HSE} 
\lfoot{\tiny Issued \firstdate} 
\cfoot{-- \thepage \ -- } \rfoot{\tiny  \sc\version}
\rhead{{\tiny  Misha Verbitsky}}


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\listok{5}{Symplectic handout 5: Hausdorff metric on convex sets}

\definition
Let $M$ be a metric space, and $X\subset M$ a subset.
The {\bf $\epsilon$-neighbourhood} of $X$ is
$X(\epsilon):= \bigcup_{x\in X} B_\epsilon(x)$,
where $B_\epsilon(x)$ is an $\epsilon$-ball centered in $x\in X$.
{\bf the Hausdorff distance} $d_H(X,Y)$ is the infimum
of all $\epsilon$ such that $X(\epsilon)\supset Y$
and $Y(\epsilon) \supset X$.
\ed

\definition
{\bf Diameter} $\diam$ of a metric space $M$ is 
supremum $d(x,y)$ for all $x, y\in M$.
\ed

\exercise
Let $M$ be a metric space of finite diameter,
and ${\goth U}$ the set of all closed subsets of $M$.
Prove that $d_H$ defines a metric on ${\goth U}$.
\ez

\exercise
Let $X_i$ be a $d_H$-Cauchy sequence of closed
subsets of a metric space $M$, and $X$ its limit.
\enum
\ite Prove that $X= \bigcap X_i(\epsilon_i)$,
for an appropriate sequence 
$\{\epsilon_i\in \R^{>0}\}$ converging to 0. 
\ite Let $\{x_i\in X_i\}$ be any sequence,
and $x$ its limit. Prove that $x\in X$.
\ite Let $x\in X$. Prove that $x$ is a limit
of a sequence $\{x_i\in X_i\}$.
\ite Prove that $X$ is the set of all
limiting points $\lim_{i \to \infty} x_i$,
for al sequences $\{x_i\in X_i\}$ 
\ite Prove that the topology on ${\goth U}$
induced from $d_H$ is determined by the
topology on $M$.
\ee
\ez

\exercise
Let $M$ be a complete metric space of finite diameter, and
${\goth U}$ the set of all closed subsets of $M$,
equipped with $d_H$-topology. Prove that
${\goth U}$ is complete.
\ez

\exercise
Let $M$ be a compact metric space, and
${\goth U}$ the set of all closed subsets of $M$,
equipped with $d_H$-topology. Prove that
${\goth U}$ is compact.
\ez

\definition
Let $X, Y$ be metric spaces.
{\bf Uniform topology} on the space
$\Map (X, Y)$ of continuous maps is 
the topology induced by the metric
\[ d(f, g):= \sup_{x\in X} d(f(x), g(x)).\]
$C^0$-topology is uniform convergence on compacts,
and $C^i$-topology is uniform convergence on compacts with
all derivatives up to $i$-th.
\ed

\exercise
Let $X, Y$ be compact metric spaces,
and $f, g \in \Map (X, Y)$.
Define $d_H(f, g)$ as the Hausdorff distance
between the graphs of $f$ and $g$. Prove that
the topology induced by this metric on
$\Map (X, Y)$ coincides with the uniform topology.
\ez

\definition
A {\bf convex hull} of $U\subset \R^n$ is the
smallest convex set containing $U$.
{\bf A simplex} is a convex hull of $n+1$ points.
\ed

\exercise
Let $\hat U\subset \R^n$ be
a convex hull of $U\subset \R^n$.
Prove that $\hat U$ is a union of
all simplices with vertices in $U$.
\ez

\exercise
Let $A\subset \R^n$ be a compact,
convex subset of $\R^n$. A point
$x\in A$ is called {\bf extremal}
if $x\neq t y + (1-t) z$
for any $t \in ]0,1[$ and any
$y, z\in A$. Prove that $A$
is a convex hull of the set
of its extremal points.
\ez

\remark
From now on, we consider $\R^n$ as a metric space
with the standard Euclidean metric.
\er


\exercise
Let ${\goth C}$
be the set of all convex, 
compact subsets of $\R^n$,
Prove that $({\goth C}, d_H)$
is complete.
\ez

\exercise
Let ${\goth O}$ be the set of all open, bounded 
convex subsets, and let
$d_U(A, B):= d_H(\R^n \backslash A, \R^n \backslash B)$.
\enum
\ite
Prove that $d_U$ is a metric, and induces the same
topology on ${\goth O}$ as $d_H$.
\ite Prove that $d_U(A, B) \leq d_H(A,B)$.
Find an example when $d_U(A, B) < d_H(A,B)$.
\ee
\ez

\exercise
For any subset $A\subset \R^n$,
consider {\bf the boundary} $\6A:= \bar A \backslash A^\circ$,
where $\bar A$ is the closure and 
$A^\circ$ is the set of interior points of $A$.
Find compact subsets
$A, B\subset \R^n$ such that 
\enum
\ite $d_H(\6A, \6B) > d_H(A,B)$
\ite $d_H(\6A, \6B) < d_H(A,B)$.
\ite Prove that $d_H(\6A, \6B) = d_H(A,B)$
when $A, B$ are convex.
\ee
\ez

\exercise
Let $A\subset \R^n$ be an open, convex, bounded subset
containing 0. Denote by $\lambda A$ the set
$\{x\in \R^n\ \ |\ \ \lambda^{-1} x\in A\}$.
Prove that for each $\lambda \neq 1$ there exists
$\epsilon$ such that $\6A(\epsilon)\cap \6 (\lambda A)=\emptyset$.
\ez

\exercise\label{_close_then_homothety_Exercise_}
Let $A\subset \R^n$  be an open, convex, bounded subset
containing 0.
Prove that for each $\delta >0$
there exist $\epsilon >0$ such that
for any convex $B$ with $d_H(B, A) \leq \epsilon$,
one has $(1-\delta) A \subset B \subset (1+\delta) A$.
\ez

\hint Use the previous exercise.
\eh

\exercise
Let $\phi:\; \R^n \arrow \R^n$ be a smooth diffeomorphism,
$\phi(0)=0$. 
Prove that there exists $\epsilon >0$ such that
$\phi$ maps an open ball $B_\delta(0)$ to a convex
set, for any $\delta < \epsilon$.
\ez

\exercise Let $\phi_i:\;  \R^n \arrow \R^n$
be a sequence of diffeomorphisms which converges
in $C^2$-topology to a diffeomorphism $\phi$.
Suppose that all $\phi_i$ and $\phi$ map 0 to 0.
Prove that for a sufficiently small Euclidean 
open ball $E\ni 0$, almost all images 
$\phi_i(E)$ are convex.
\ez

\definition
Let ${\goth O}$ be the set of all open, bounded 
convex subsets.
A function $c:\; {\goth O} \arrow \R^{\geq 0}$
is called {\bf convex capacity} if it 
is invariant under isometries and
satisfies $c(\lambda E) = \lambda^2 c(E)$ and
$c(E_1) \geq c(E_2)$ whenever $E_1 \supset E_2$.
\ed

\exercise
Prove that any convex capacity is continuous
on ${\goth O}$ in the topology defined by the
Hausdorff metric.
\ez

\hint Use 
Exercise \ref{_close_then_homothety_Exercise_}. \eh

\exercise
Let $\phi_i:\;  \R^n \arrow \R^n$
be a sequence of diffeomorphisms which converge
in uniform topology to a diffeomorphism $\phi$.
Prove that for any compact set $E\subset \R^n$,
$\phi_i(E)$ converges to $\phi(E)$ in the
topology given by the Hausdorff metric.
\ez

\exercise
Let $\phi_i:\;  \R^n \arrow \R^n$
be a sequence of diffeomorphisms which converge
in uniform topology to a diffeomorphism $\phi$,
and $c$ a convex capacity. Suppose that
$\phi_i$ and $\phi$ map a given convex subset $E\in {\goth O}$
to a convex subset. Prove that $\lim_i c(\phi_i(E)) = c(\phi(E))$.
\ez

\exercise
Let $\phi:\; \R^n \arrow \R^n$
be a diffeomorphism, and $c:\; {\goth O} \arrow \R^{>0}$ 
a convex capacity. Suppose that $c(\phi(E))= c(E)$
for any $E\in {\goth O}$ such that $\phi(E)$ is convex.
Let $h_\lambda(x)=\lambda x$,
and let $\phi_\lambda(z) := h_\lambda(\phi(h_\lambda^{-1}(z)))$.
\enum
\ite
Prove that $\phi_\lambda$ converges to the differential $d\phi$
uniformly on compacts as $\lambda$ goes to $\infty$.
\ite Prove that $c(d\phi(E))= c(E)$ for any ellipsoid
$E\subset \R^n$.
\ee
\ez








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