# 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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### 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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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### 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 ...
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### 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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### 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 ...
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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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### 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). ...
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