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Consider the following property of the Hausdorff distance in $\mathbb R^n$.

Let $\Omega_n \supset \Omega_{n+1} \supset ...$ a sequence of open, convex and bounded sets with $\operatorname{int}(\bigcap \Omega_n) \neq \emptyset$. Then $\Omega_n \rightarrow \operatorname{int}(\bigcap \Omega_n)$ in the Hausdorff distance.

Times ago I found on the internet the proof of the above property in a lecture note. But now the site does not exists and I am not finding a book with these properties. Someone could point me a reference with the properties above?

Days ago I asked this at math stack exchange, but no one answered... I dont know if here is the correct place to ask, if is not the right place, sorry.

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    $\begingroup$ On such occasions I used to write: it is well known and easy to see that [...]. $\endgroup$
    – Włodzimierz Holsztyński
    Commented Sep 8, 2014 at 17:38
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    $\begingroup$ @WlodzimierzHolsztynski And also "it follows readily from the definition that..." $\endgroup$
    – Vidit Nanda
    Commented Sep 8, 2014 at 19:26
  • $\begingroup$ Leandro, you're too kind :-) $\endgroup$
    – Włodzimierz Holsztyński
    Commented Sep 8, 2014 at 22:13

2 Answers 2

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You could find a proof in Variation et optimisation des formes by Antoine Henrot and Michel Pierre, Section 2.2.3.

The second property mentioned there is the following:

An increasing sequence $(K_n)$ of compacts included in a compact box $B$ converges towards the closure of its union.

Taking complements you arrive exactly where you want.

Proof: Denote $K = \overline{\cup K_n}$, and consider $x_n \in K$ such that $d(x_n,K_n)=\sup_{x \in K}d(x,K_n)=d_H(K,K_n)$ (here we use the inclusion). Pick $x$ a limit point of $x_n$ in $K$ (here we use the compacity of $K$). Then $d(x_n,K)\leq d(x_n,x)\to 0$, which means that $d_H(K,K_n) \to 0$.

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I believe this follows easily from the Blaschke Selection Theorem.

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