Questions tagged [hausdorff-distance]

For questions involving the Hausdorff distance (also known as the Hausdorff metric) between closed, bounded, non-empty subsets of normed linear spaces (or more generally, any metric space).

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Convergence in Hausdorff distance $\iff$ Lebesgue measure converges

There are numerous sources (e.g. here) mentioning that Lebesgue measure is not continuous with respect to the Hausdorff distance $d_H$. This means that for a sequence of sets $K_n$ converging to $K$, $...
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About a classic explanation of $\mathcal A=\pi R^2$ for a disk

In a recent post, OP deals with a well known explanation of the formula $$\mathcal A=\pi R^2$$once you know $$p=2\pi R$$ When this explanation is given, it is usually sufficient to say that the areas ...
Stéphane Jaouen's user avatar
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Seperability for the collections of all non-empty compact subsets of $\mathbb{R}^2$ with Hasudorff metric

Let $X$ be the collections of all non-empty compact subsets of $\mathbb{R}^2$, which has the Euclidean metric. Let $(X,d)$ be a metric space, where $d$ is the Hasudorff metric. Is $X$ separable? ...
Paul H.Y. Cheung's user avatar
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Is subset relation preserved under limit for Hausdorff metric?

Let $X$ be a metric space. I consider elements in $Y=2^X\setminus \emptyset$ and use the Hausdorff metric for $Y$. Suppose that $A_n \subseteq B_n$ for $A_n,B_n \in Y$ and $A_n \rightarrow A$ and $B_n ...
Paul H.Y. Cheung's user avatar
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Largest circle contained within a triangle

I am interested in figures of elementary geometry that can be used to illustrate the Hausdorff distance. You don't even have to know what it is. Here's what it looks like in geometrical terms: Let $T=...
Stéphane Jaouen's user avatar
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Show convergence of sets

Consider the following sets: $$ \begin{aligned} & A_n\equiv \Big\{ x\in X: d\big(p_n, [\ell(x), u(x) ] \big)\leq \delta_n\Big\}\\ & A \equiv \Big\{ x\in X: \lim_{n\rightarrow \infty} d\...
Star's user avatar
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Farther and nearest points from an ellipse to the a line segment.

I really need your help in math, I've solved this problem in several ways, but I'm not getting anywhere... Given a straight line $x-3y-9=0$ and an ellipse $x^2/9 +y^2/4 = 1$, find the nearest and ...
Masha's user avatar
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Hausdorff Distance Confusion

I was confused about something really basic about Hausdorff distances and was wondering if someone could clarify. Let $A \subset X $ be a non-empty set, and $a \in A$. Suppose further that $x \in X$. ...
Sean Thrasher's user avatar
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Lipschitz function and Hausdorff distance

I am wondering if we can assume the following if your function $f$ is $K-$lipschitz: $$\delta(f(A),f(B)) \leq K*\delta(A,B),$$ where we use the definition: $$\delta(A,B) = \inf\{\epsilon \geq 0\,|\ A \...
Frederick Manfred's user avatar
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Is the Lipschitz constant of the Hutchinson operator the maximum of the Lipschitz constant of its components?

Let $(X,d)$ be a metric space, define $F(X)$ as the set of all non-empty compact subsets of $X$, for $A,B \in F(X)$ we define $$d(A,B) = \sup_{a \in A} \, \inf_{b \in B} d(a,b)$$ We now define the ...
H. de Gracht's user avatar
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Quasi-geodesic rays are closed to geodesic rays in proper hyperbolic geodesic spaces

We define the boundary of a hyperbolic metric space $\partial X$ as the equivalence classes of geodesic rays up to finite Hausdorff distance and $\partial_q X$ as the equivalence classes of quasi-...
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Question about proof of Krantz proposition 1.6.14 (Steiner symmetrization preserves compacteness)

I was reading the introductary chapter on the book Geometric integration theory (Birkhäuser, Cornerstones series 2008, Krantz & Parks) and stumbled upon the following theorem/proposition about ...
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Steiner symmetrization preserves compactness?

I have read " Convexity, H.G.Eggleston, 1958'' and in page 91, theorem 43, it proves that a closed, bounded and convex set $\mathcal{X}$ is still closed bounded (and convex) under Steiner ...
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Compact set of "symmetric difference metric space"

Problem: Show that $I(A)$ is not compact in $(X, \delta)$. Let $X$ be a set of convex polygons in a plane, and for all $A \in X$, $m(A)$ denotes the area of $A$. We define a distance function on $X$ ...
T. Sen's user avatar
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Subset of continuous functions from [0,1] to R closed for the Hausdorff metric

For the past few days I've been studying metric spaces and I was just making some exercices since I have an exam on it in a few days. I was struggling a bit with the following problem: let $G$ be the ...
luki luk's user avatar
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In the space $K(X)$ with the topology induced by the Hausdorff metric, $K_f(X)=\{K\in K(X): K\,\text{is finite}\}$ is $F_{\sigma}$.

The Problem: Let $X$ be metrizable, let $K(X)$ be the space of all compact subsets of $X$ with the topology induced by the Hausdorff metric $d_H$. Show that the set $K_f(X)=\{A\in K(X): K\,\text{is ...
Dick Grayson's user avatar
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V a IPS of finite dimension, $\phi : V \to F$ is a linear functional. Let $B = \{v_1...v_n\}$ an orthonormal basis for V.

Prove Riesz unique representation theorom: If $v \in V$ is a vector in $V \implies \exists ! u \in V$ that stasfies $\phi (v) = \;<v,u>$ $\mathbf {Hint}$: express $u$ as a linear combination of ...
MathStudent101's user avatar
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Hausdorff Distance Between Orthogonal Complements

Let $H$ be a finite-dimensional complex Hilbert space and denote by $d_{\textrm{Haus}}$ the Hausdorff distance between linear subspaces of $H$ i.e., $d_{\textrm{Haus}}(V,W)$ is the usual Hausdorff ...
gm01's user avatar
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How to solve exercise 4.15 in stein functional analysis?

In this problem, the Hansdorff distance of two sets $A,B$ is $dist(A,B)=\inf\{\delta:A^{\delta}\subset B,B^{\delta}\subset A\}$, in which $A^{\delta}=\{x:dist(x,A)<\delta\}$ One of my ideas is to ...
呆呆唯's user avatar
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Counting fractals via "stagewise complements"

This is motivated by this older question. Now posted at MO. Let $\mathscr{H}$ be the space of compact nonempty subsets of $\mathbb{R}^2$ (I'm not especially wedded to dimension $2$, so feel free to ...
Noah Schweber's user avatar
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Find $n$ points that minimize the supremum distance between any point and the closest point in the set

Let $(\mathcal{X}, d)$ be a metric space and $n \in \mathbb{N}$. Find $$\operatorname*{argmin}_{\mathcal{Y} \in [\mathcal{X}]^n} \sup_{x \in \mathcal{X}} \inf_{y \in \mathcal{Y}} d(x, y)$$ That is, ...
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Prove that a decreasing sequence of closed and bounded sets converges in the Hausdorff metric

Let $(X,d)$ a complete metric space. I need to prove that $(\mathcal{CB}(X),d_H)$ is complete. I see other posts (yes, it's a duplicate but no one answered this question) and the only thing I couldn't ...
Raúl Filigrana Villalba's user avatar
4 votes
3 answers
144 views

Is there a metric that makes $\mathbb{N} \cup \{\infty\}$ a complete metric space?

Denote the set of extended natural numbers $\mathbb{N} \cup \{\infty \}$ by $X$. Is there any metric $D$ that makes $X$ a complete metric space? In other words a distance function $D: \mathbb{N}\times ...
Elias Costa's user avatar
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Metric spaces and ultrafilters

Assume $X_n$ are proper metric spaces and $X$ is a compact metric space. Fix an non-principal ultrafilter $\omega$ on $\mathbb{N}$. Assume we have a metric space $\mathbb{X}$ such that: (1) $\mathbb{X}...
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If $x_n \to x$ then $d_H(\partial h(x_n),\partial h(x)) \to 0$

Let $X := \mathbb R^d$ and $h: X \to \mathbb R$ be convex. Then $h$ is locally Lipschitz, and $\partial h (x)$ is non-empty, convex, and compact for all $x \in X$. Let $d_H$ be the Hausdorff ...
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The inverse of the subdifferential of a strictly convex function is Borel measurable

Let $X := \mathbb R^d$ and $h: X \to \mathbb R$ be strictly convex. Then $\partial h (x)$ is non-empty, convex, and compact for all $x \in X$. $\partial h (x) \cap \partial h (y) \neq \emptyset$ if ...
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Why do the researchers usually study the collection of **nonempty** closed subsets?

In the research field of hypertopologies, researchers usually study on nonempty closed subsets of a topological space, although I have seen, but a very few books or articles, where the authors have ...
ask's user avatar
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Is "Escherian metamorphosis" always possible?

This question is motivated by Escher's series of Metamorphosis woodcuts (see e.g. here), where one tesselating tile is gradually transformed into another. Basically, this is a precise way of asking ...
Noah Schweber's user avatar
5 votes
1 answer
229 views

Is Gromov-Hausdorff distance realized when one space is compact?

The Gromov-Hausdorff distance between two complete$^1$ metric spaces $M,N$ is defined as the infimum, over all pairs $f,g$ of embeddings of $M,N$ into some third metric space $U$, of the Hausdorff ...
Noah Schweber's user avatar
2 votes
0 answers
88 views

Is there a characterization of convergence in Hausdorff metric in terms of that in the underlying metric?

Let $(E, d)$ be a metric space and $\mathcal C$ the set of all non-empty compact subsets of $E$. We endow $\mathcal C$ with the Hausdorff metric $d_H$. It's well-known that the topology of $\mathcal C$...
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Let $A_n \to A$ in $d_H$, $\mu_n \to \mu$ weakly, and $\operatorname{supp} \mu_n \subset A_n$. Then $\operatorname{supp} \mu \subset A$

Let $(E, d)$ be a metric space and $\mathcal C$ the set of all non-empty compact subsets of $E$. We define the Hausdorff metric $d_H$ on $\mathcal C$ by $$ d_H (A, B) := \max \left \{ \max_{x \in B} ...
Analyst's user avatar
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If the underlying metric is totally bounded, then so is its induced Hausdorff metric

Let $(E, d)$ be a metric space and $\mathcal C$ the set of all non-empty compact subsets of $E$. We define a metric $d_H$ on $\mathcal C$ by $$ d_H (A, B) := \max \left \{ \max_{x \in B} d(x, A), \...
Analyst's user avatar
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If the underlying metric is complete, then so is its induced Hausdorff metric

Let $(E, d)$ be a metric space and $\mathcal C$ the set of all non-empty compact subsets of $E$. We define a metric $d_H$ on $\mathcal C$ by $$ d_H (A, B) := \max \left \{ \max_{x \in B} d(x, A), \...
Analyst's user avatar
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Does the supremum norm $\|p\|_{A}$ depend continuously on subsets $A\subset\mathbb{C}$ with respect to the Hausdorff distance?

Consider the space $\mathcal{K}$ of all non-empty compact subsets of $\mathbb{C}$. One can show that the Hausdorff distance defined by $$h(X,Y)=\max\bigg\{\sup_{x\in X}\inf_{y\in Y}|x-y|,\sup_{y\in Y}\...
Calculix's user avatar
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Hausdorff distance on the metric space containing compact subsets

I'm trying to verify that $d_H$ is indeed a bona fide metric. Could you have a check on the triangle inequality part? Theorem: Let $(E, d)$ be a metric space and $\mathcal C$ the set of all compact ...
Analyst's user avatar
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Approximating a Compact set by approximating its distance function

Let $\emptyset\neq K\subset Y$ be a closed subset of a compact metric space $(X,d)$ such that $K$ has at-least two points and such that $$ d(Y,K):=\sup_{y\in Y}\,\inf_{k\in K}\,d(k,y)=:r>0. $$ ...
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Is the following result on the Hausdorff distance even true?

In this post on the Hausdorff distance $d_H$, the following is stated as a Theorem: Theorem: Let $A,B,C \in H(X)$ (where $H(X)$ is the set of non-empty compact subsets of $X$). Then $d_H(A \cup B,C) ...
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Convergence of superlevel sets to the argmax

Let $f:C\mapsto[-\infty,\infty]$ be an extended real valued function, where $C \subset \mathbb{R}^n$, and let $lev_\alpha f=\{x: f(x) \geq \alpha\}$ be the corresponding $\alpha$-superlevel set. ...
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The range of Hausdorff distance values

What can we interpret if Hausdorff distance between 2 sets is negative or positive? I found that Hausdorff distance equal to zero implies the 2 sets are very close to each other. Also Is Hausdorff ...
Shivanisrivarshini's user avatar
2 votes
1 answer
490 views

When does convergence in Hausdorff distance imply convergence in measure of indicators?

Given a sequence of compact subsets of the real line which converges in the Hausdorff metric, it is not guaranteed that the sequence of indicator functions of the given subsets converge in measure (...
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Sets known with Hausdorff distance

Let A and B be two arbitrary compact sets. Then what property should hold by A and B such that the distance D(A, B) = D(B, A) = h(A, B), where D is the set distance and h is the Hausdorff distance. ...
Kifayat Ullah Lone's user avatar
4 votes
1 answer
133 views

Is there a normed vector space of shapes?

I've recently been interested in the Hausdorff Distance (a notion of distance between sets/shapes), and I am curious if there is a normed (or semi-normed) vector space of sets in $\mathbb{R}^n$, where ...
Zim's user avatar
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Show that if $f_i: X \to X$ are contractions on a metric space $X$, then $F(K)=\bigcup_{i=1}^N f_i(K)$ is a contraction on $\mathcal{H}(X)$.

Show that if $f_i: X \to X$ are contractions, then $F: \mathcal{H}(X) \to \mathcal{H}(X)$ defined by $F(K)=\bigcup_{i=1}^N f_i(K)$ is a contraction on $\mathcal{H}(X)$, where $\mathcal{H}(X)$ is the ...
Dick Grayson's user avatar
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Prove that the Hausdorff Distance satisfies the Triangle Inequality.

Similar questions have been asked here and here. Let $\mathbb{H}(X)$ denote the set of all nonempty compact subsets of $X$, where $(X, d)$ is a metric space. Define the Hausdorff Distance as $d_{\...
Dick Grayson's user avatar
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How to show the limit space is totally bounded in the proof of completeness of Gromov-Hausdorff metric space?

The Gromov-Hausdorff metric space $(\mathcal{M},d_{GH})$ is complete. I'm currently following the proof of this fact given in Petersen's Riemannian Geometry (3rd Edition) (see Proposition 11.1.8). ...
Hopf eccentric's user avatar
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1 answer
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Understanding details of Bishop-Peres' proof of Lemma A.23. If $(X,d)$ is complete, thent $(\mathrm{Cpt}(X),d_H)$ is also complete.

In the following book, page 338, there is Lemma A.23: If $(X,d)$ is a complete metric space, then $(\mathrm{Cpt}(X),d_H)$ is a complete metric space. I want to understand some details: (i) Why is $K_n ...
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Uniform convergence over a moving interval

Let $f,f_n : [0,8] \to \mathbb{R}$ two continuisly differentiable functions defined over $[0,8]$ for all $n \geq 1$. If $f_n$ uniformly converges to $f$ over $[4,5]$ when $n\to +\infty$. and $f_n$ ...
BrianTag's user avatar
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2 votes
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Gromov–Hausdorff distance between a single point and a nonempty compact subset K of a metric space is equal to half the diameter of K

I need to show that the Gromov–Hausdorff distance between a single point and a nonempty compact subset $K$ of a metric space is equal to half the diameter of $K$, and I don't see how to do this. In ...
juan1243's user avatar
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2 answers
434 views

Hausdorff distance between convex sets in $\mathbb{R}^d$.

I have the following property I suspect about the Hausdorff distance between convex sets in $\mathbb{R}^d$, and I thought that perhaps someone can find a fault with this idea if it is wrong. Given ...
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What's the distance between triangles?

How can we define distance between, say, two of them? And if we the distance between two of them and the distance of another one to the first, do we know its distance tö rhe second? What ways are ...
Deschele Schilder's user avatar