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Questions tagged [hausdorff-distance]

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Intersection in the Hausdorff metric space

Let $(X,d)$ and $(Y,d)$ be complete metric space, and denote by $H(Y,d)$ the Hausdorff metric space on the compact subsets of $(Y,d)$. Let $$ f,g:(X,d)\rightarrow H(y,d), $$ be continuous maps (may ...
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Show when $x\neq y$ then in a Metric space $d(x,y):= \epsilon > 0$ [closed]

Let $X$ be a Metric space and $x\neq y$. Show that $d(x,y):= \epsilon > 0$ and I am given as a hint to show that that $B_{ \frac{\epsilon}{2}}(x)\cap B_{ \frac{\epsilon}{2}}(y)=\varnothing$ My ...
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Hausdorff metric and connectedness [duplicate]

Let $(X, d)$ be metric space. Define $B_\epsilon = \{ x \in X : \exists b \in B \; d(x, b) \le \epsilon\} $. Let $F(X)$ be a family of all nonempty compact subsets of $X$ (so $\emptyset \notin F(X)$ )....
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$\epsilon$ fattening of an open set needn't be open

This was a question on a test. Let $(X,d)$ be a metric space and $\epsilon \geq 0$. Define the $\epsilon$-fattening of a set $S$ as $$S_\epsilon:=\{ x \in X; \exists s \in S: d(s,x) \leq \epsilon \}$...
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What does the reparameterization mean in Fréchet distances?

I am trying to understand the definition of frechet distance but I am struggling to understand the reparameterization part in the definition. I got the following definition from wikipedia Let A and ...
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29 views

Distance from a point to a curve

Given a continuous map $g:[0,1]\longrightarrow \mathbb{R}^{n}$, Somebody know an efficient algorithm to compute the (say, the Euclidean) distance from a point $P\in\mathbb{R}^{n}$ to $g(I)$? Of ...
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Frechet distance decomposability- Can we do a divide and conquer approach for Frechet?

Given a dataset of Polygonal Lines which are partitioned using equal-sized grid cells. Suppose two Lines P and Q which are segmented at cell boundaries and each of which is converted to two ...
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Encoding the position information in real distance metric

The Euclidean distance doesn't preserve the exact position information. For example, the distance of the points (3,1) and (1,3) would be the same from the origin. Is there any distance metric which ...
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Hausdorff partial metric

Using a metric on a metric space $(X,d)$ we can define Hausdorff metric $h$ on the set of all compact subsets of $(X,d)$,say $K(X)$. I am able to show that $h$ is indeed a metric. Suppose instead of ...
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54 views

A bound for the Hausdorff distance

Let $(X,d)$ be a compact metric space and $K(X)$ the class of its compact subsets. Recall, that the Hausdorf distance between $A,B\in K(X)$ is given by $$ d_{H}(A,B):=\max\big\{ \sup_{x\in A}\inf_{y\...
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Proving a result about Hausdorff Distance

Let $(M,d)$ be a metric space. The Hausdorff distance between two nonempty sets $S,T\subseteq M$ is defined by $$ d_H(S,T):=\max(\{\sup\big(\{\tilde{d}_H(a,T):a\in S\}\big), \sup\big(\{\tilde{d}...
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Equivalent geodesic rays

I am reading the book 'Metric spaces of non-positive curvature' by Bridson and Haefliger. In page 427 the following is said: Two geodesic rays $c, c^\prime \colon [0, \infty)\rightarrow X$ are said ...
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Kernel function from Hausdorff distance

Is there a way to build a positive definite kernel on the space of arbitrary point-sets in $\mathbb{R}^d$ using the Hausdorff distance? I tried the obvious option of setting $K(X, Y) = \exp\{-d_H(X, Y)...
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A way to prove that Hausdorff distance is complete

I have to solve an exercise the result of which will lead to the completeness of the Hausdorff distance. Basically it si the fulfilling of the details of this answer https://math.stackexchange.com/q/...
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Some questions about Hausdorff distance

First of all the definitions I am working with: In all the following $ (X,d) $ will be a metric space and $B(x,r) = \{ y \in X : d(x,y)<r \}$ for any $x \in X$ and $r>0$. .Let $C \subset X$ ...
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Continuity of the kernel of bounded operators under perturbation

In a nutshell: Does the kernel of a bounded operator change "nicely" with the operator? The details: Let $(X,\| \|)$ be an infinite-dimensional real normed space. Let $A_t $ be a continuous family ...
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Convergence properties in a Hausdorff space that is a ring

Suppose $T$ is a Hausdorff space and that $T$ forms a ring with identity. Is the following claim true? If $lim_{n\to \infty}a_n=0$, Then $lim_{n\to \infty}ra_n=0$ . (where $a_n \in T$ and $r ...
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How to properly read Hausdorff Metric?

How can I read and understand the formula of Hausdorff metric? I understand the concept, but I am not sure what this formula really wants to tell me? I have seen an explanation from this link that, ...
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Hausdorff distance, Translation and Intersection

For two subsets $X,Y$ of a metric space $(M,d)$ the Hausdorff distance is defined as $$ d_{\mathrm H}(X,Y) = \max\{\,\sup_{x \in X} \inf_{y \in Y} d(x,y),\, \sup_{y \in Y} \inf_{x \in X} d(x,y)\,\},$$ ...
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Hausdorff distance between two set of finite points

Assume we have two (finite) sets of points, $A:=\{a_{1},\ldots,a_{n}\}$ and $B:=\{b_{1}, \ldots,b_{n}\}$ in the closed hypercube $[0,1]^{d}\subset \mathbb{R}^{d}$. Somebody know an "efficient" ...
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Bounds on the difference of sups in terms of the Hausdorff distance between feasible regions

Let $f: \mathbb{R}^{k} \mapsto \mathbb{R}$ be a bounded and continuous function, and suppose that $X \subset \mathbb{R}^{k}$ and $Y \subset \mathbb{R}^{k}$ are compact. I am reading a paper that ...
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Locally compact hausdorff property.

$X$ is Hausdorff space. The following is equivalent. (a) $X$ is locally compact space. (b) For every open neighborhood $U$ of $x\in X$ there is a smaller open neighborhood $V$ of $x$ whose closure ...
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A set is bounded for the Hausdorff distance iff the union of all of its members is bounded.

Let $E = \mathbb{R}^N$ for some $N \in \mathbb{N}$ and $d(x,y)=||x-y||_2$ the euclidean distance. We note $K(E)$ the set of all compact sets of $E$, and we'll use the Hausdorff distance on $K(E)$ ...
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Continuity of Lebesgue measure w.r.t Hausdorff distance? [duplicate]

If one considers the metric space $\mathcal{K}$ of all compact subsets of $\mathbb{R}^2$ endowed with the Hausdorff distance (that is $\Delta(A,B)=\inf \{ \delta: A\subset B^{\delta},B \subset A^{\...
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Get the maximum diverse of subset without compute all distance points

I read slide about top-k algorithms which are Fagin Algorithm (FA), Threshold Algorithm (TA) and No random access algorithm (NRA). Those algorithms are good because we do not need to see all values, ...
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Convex hulls change continuously as one point moves continuously

I am seeking a reference or a succinct proof of this claim: Let $H$ be the convex hull of a finite set of points $p_1,p_2,\ldots$ in $\mathbb{R}^d$. Move one point $p_1$ along a continuous rectifiable ...
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174 views

Union of a Cauchy sequence of Compact Sets

Let $X$ be a complete metric space. Define $H$ to be the set of non-empty compact subsets of $X$. Now let $A_n$ be a cauchy sequence in $H$. (Considering the Hausdorff metric on $H$). Then can it be ...
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Hausdorff distance between two convex bodies

I consider all the convex bodies of constant width in $\mathbb{R^2}$ (to fix ideas I take the width equal 1 ) I consider that all these bodies are centered in O( O is the Origine the Cartesian ...
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267 views

Haussdorff convergence in $\mathbb{R}^n$

Let $\omega_n$ be bounded open subsets of $\mathbb{R}^n$ such that $(\omega_n)$ converges to $\omega$ in sens of Hausdorff metric. I would like to know what are the boundary conditions, if there ...
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Convergence of sets on a Hausdorff space (looking for a published reference)

I need a reference for the following facts: that on $T_1$ spaces, limit and cluster points coincide that if $(E,d)$ is a compact metric space, then $(\mathcal{K}(E),d_H)$ is also a compact metric ...
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How to find Hausdorff Distance between $B$, $C$

Let \begin{align} A &= \{(x,y) \in \mathcal{R^2} : - 1 \leq x \leq 1 \land -1 \leq y \leq 1\}, \\ B &= \{(x,y) \in \mathcal{R^2} : x^2 + y^2 = 1\}, \text{ and} \\ C &= \{(x,y) \in A : |...