13.3 Hausdorff Distance
Lemma 13.3.1.label Let $C\suf\R^{n}$ be a nonempty closed convex set and $C\suf\cl{B}(0,R)$ for some $R>0$ then for all $u,v\in\R^{n}$
In particular, $h_{C}|_{\mathbb{S}^{n-1}}$ is $R$-Lipschitz.
Proof. For every $x\in C$
Taking maximum over $x\in C$ gives $h_{C}(u)\leq h_{C}(v)+R\norm{u-v}{}$. Interchanging $u$ and $v$ gives the result.$\square$
Definition 13.3.2 (Hausdorff Distance).label Let $A\suf\R^{n}$ be a nonempty set and $x\in\R^{n}$ then
When $A$ is compact, the infimum is attained. For compact $A$, the closed $\e$-neighborhood of $A$ is $A_{\e}\define A+\e B^{n}_{2}=\curl{x:d(x,A)\leq \e}$ where $B_{2}^{n}=\curl{x\in\R^n:\norm{x}{2}\leq 1}$ is the Euclidean unit ball.
Let $C,D\suf\R^{n}$ be nonempty compact sets. Their Hausdorff distance is
If in addition $C,D$ are convex then
Proof. It suffices to show $d_{H}(C,D)\leq \e\iff C\suf D+\e B^{n}_{2},D\suf C+\e B^{n}_{2}$. But $\SUP{x\in D}d(x,D)\leq\e$ is equivalent to $C\suf D+\e B^{n}_{2}$.$\square$
Theorem 13.3.3 (Blaschke Selection Theorem).label Let $\parens{K_j}_{j\in\N^+}$ be a sequence of nonempty compact convex sets in $\R^{n}$ such that $\exists R<\infty$ with $K_{j}\suf RB^{n}_{2}$ for every $j\in\N^{+}$. Then $\exists j_{\el}\suf\N$ and nonempty convex set $K\suf RB^{n}_{2}$ such that $d_{H}(K_{j_\el},K)\to 0$.
Proof. We write $h_{j}\define h_{K_j}:\mathbb{S}^{n-1}\to\R$ then $\abs{h_j(u)}\leq R$ for each $j$ by $K_{j}\suf RB^{n}_{2}$. By Lemma 13.3.1, $\curl{h_j}_{j}$ is uniformly bounded and equivcontinuous on the compact set $\mathbb{S}^{n-1}$. By Arzela-Ascoli, after passsing to a subsequence there exists a continuous function $h:\mathbb{S}^{n-1}\to\R$ such that
Define
which is closed and convex as an interesection of closed half-spaces. Let $y_{j_\el}\in K_{j_\el}\suf \cl{B}(0,R)$ then $\inner{y_{j_\el}}{u}\leq h_{j_\el}(u),\fall j_{\el},\fall u\in \mathbb{S}^{n-1}$ then there exists a convergent subsequence $\curl{y_{j_\el}}_{\el\geq 1}$ denoted the same way for simplicity such that $y_{j_\el}\to y\in \cl{B}(0,R)$ then $\inner{\cd}{u}$ is bi-continuous so $y\in K$. Let $z\in K$ and set $u\define\frac{z}{\norm{z}{}}$ then $\inner{z}{u}=\norm{z}{}\implies \norm{z}{}\leq h(u)\leq R\implies K\suf\cl{B}(0,R)$ hence $K$ is compact nonempty. Clearly $h_{K}(u)\leq h(u)$ for all $u\in \mathbb{S}^{n-1}$. Fix $u\in \mathbb{S}^{n-1}$ then $\fall j_{\el},\exists y_{j_\el\in K_{j_\el}}$ such that $\inner{y_{j_\el}}{u}=h_{j_\el}(u)$ so there exists $y\in \cl{B}(0,R)$ such that $y_{j_\el}\to y$ so $\inner{y}{u}=\limit{j_\el}\inner{y_{j_\el}}{u}=\limit{j_\el}h_{j_\el}(u)=h(u)$ hence $h_{K}(u)=\MAX{x\in K}\inner{x}{u}\geq h(u)$ for any $u\in \mathbb{S }^{n-1}$ showing $h_{K}=h$ is the support function of $K$ and $K_{j_\el}\to K$ in Hausdorff distance.
It remains to show next time $K\neq\emp,K$ is compact and $h_{K}=h$ hence $K_{j_\el}\to K$ in Hausdorff distance.$\square$
Remark 13.3.4 (Isoparametric Problems).label If $A\suf\R^{n}$ compact convex then
with equality if and only if $A=B_{2}^{n}$. In $n=2$ for $\gam$ a smooth closed curve enclosing the domain $D$, this reduces to $P(\gam)^{2}\geq 4\pi Area(D)$.
Theorem 13.3.3 shows that Steiner symmetrization exists for the class of nonempty compact convex sets in $\R^{n}$.
Post a Comment