Hausdorff Metric: Definition 1 vs. Definition 2

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In summary, the article discusses the concept of the Hausdorff metric and the induced Hausdorff metric space, defined as the set of nonempty closed bounded subsets of a metric space. While the term "Hausdorff space" typically refers to a topological separation axiom, in this paper it refers to a metric space. The question is raised of whether using a different definition of the induced Hausdorff space, based on nonempty compact subsets, would yield the same results and properties. However, this would need to be proven separately, as the two definitions are not necessarily equivalent.
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Mr Davis 97
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http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.353.633&rep=rep1&type=pdf
The Hausdorff metric is defined on the space of nonempty closed bounded subsets of a metric space. The resulting metric space will be referred to as the "induced Hausdorff metric space", or else simply as the "induced Hausdorff space". Normally the term "Hausdorff space" refers to a space satisfying a certain topological separation axiom. But since this paper refers only to metric spaces,
which all satisfy the Hausdorff separation axiom, there will be no ambiguity.
<Moderator's note: Quotation added to make clear, what the author meant by "Hausdorff metric".>

The article above explores the notion of the Hausdorff metric. In the beginning of the article it describes how, if M is a metric space, the Hausdorff space induced by M is the set of all nonempty subsets of M that are closed and bounded together with the Hausdorff metric. Call this definition 1.

Here is my question: What if we instead claimed that the Hausdorff space induced M is the set of all nonempty compact subsets of M? Call this definition 2. Since the set of all nonempty compact subsets of M is a subset of the set of all nonempty subsets of M that are closed and bounded, do some of the results, which use definition 1, still follow if we use definition 2? For example, if I prove that definition 1 forms a metric space, is it immediately obvious that definition 2 does also, since it is a metric subspace of the former?
 
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Mr Davis 97 said:
Since the set of all nonempty compact subsets of M is a subset of the set of all nonempty subsets of M that are closed and bounded
M is a metric space. Why do you think that these two families of subsets are not the same?
 
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It is possible that some of the results from definition 1 may still hold for definition 2, but it would need to be proven separately. The fact that definition 2 is a subset of definition 1 does not guarantee that all the same properties and results will hold. In order to prove that definition 2 also forms a metric space, you would need to show that it satisfies all the necessary conditions for a metric space, such as the triangle inequality and the existence of a distance function. It may also be necessary to modify some of the proofs to account for the differences between definitions 1 and 2. Therefore, it is not immediately obvious that definition 2 will also form a metric space, and it would need to be proven separately.
 

1. What is the Hausdorff metric?

The Hausdorff metric is a mathematical tool used to measure the distance between two sets. It is commonly used in the field of topology to determine if two sets are similar or different.

2. What is the difference between Definition 1 and Definition 2 of the Hausdorff metric?

Definition 1 of the Hausdorff metric measures the distance between two sets by finding the greatest distance between any point in one set and its nearest point in the other set. Definition 2, on the other hand, considers all possible pairs of points in both sets and calculates the distance between them, then takes the maximum value.

3. Which definition of the Hausdorff metric is more commonly used?

Definition 1 is more commonly used as it is easier to calculate and provides the same value as Definition 2 in most cases. However, Definition 2 is preferred when dealing with more complex sets or when a more precise measurement is needed.

4. How is the Hausdorff metric used in mathematics?

The Hausdorff metric is commonly used in the field of topology to determine if two sets are similar or different. It is also used in computer vision, where it is used to compare and match images, and in data analysis to measure the similarity between data sets.

5. Are there any limitations to using the Hausdorff metric?

One limitation of the Hausdorff metric is that it is not defined for all types of sets. It also does not take into account the shape or orientation of the sets, only their overall distance. Additionally, it can be computationally expensive to calculate, especially when dealing with large or complex sets.

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