# Kantorovich Norm Explained: Pushforward Measure & Integration

• johnqwertyful
In summary, the Kantorovich norm is a mathematical concept used to measure the distance between two probability measures, named after Russian mathematician Leonid Kantorovich and commonly used in optimal transport. It is calculated by finding the supremum of the difference between two integrals, divided by the total mass that needs to be transported. The pushforward measure is used in relation to the Kantorovich norm to map one measure onto another. It has practical applications in fields such as image processing, economics, and machine learning, particularly in transportation planning. In mathematics, it is significant for its role in measuring distance between measures and its use in developing new theories and applications.
johnqwertyful
I've seen a few references to it, but can't find it defined anywhere. What is it? Something with integrating distances between measures and something. Pushforward measure or something else?

I did.

## 1. What is the Kantorovich norm?

The Kantorovich norm is a mathematical concept that measures the distance between two probability measures. It is named after the Russian mathematician Leonid Kantorovich and is commonly used in the field of optimal transport.

## 2. How is the Kantorovich norm calculated?

The Kantorovich norm is calculated by finding the supremum (or maximum) of the difference between two integrals, where one integral is taken over all possible transport plans between the two probability measures. This supremum is then divided by the total mass that needs to be transported.

## 3. What is the pushforward measure in relation to the Kantorovich norm?

The pushforward measure is a way to transform one probability measure into another by applying a function. In the context of the Kantorovich norm, the pushforward measure is used to map one probability measure onto another, allowing for the calculation of the distance between the two measures.

## 4. How is the Kantorovich norm used in practical applications?

The Kantorovich norm is used in a variety of applications, including image processing, economics, and machine learning. One common use is in transportation planning, where it can help optimize the flow of goods or people from one location to another.

## 5. What is the significance of the Kantorovich norm in mathematics?

The Kantorovich norm is significant in mathematics because it provides a way to measure the distance between two probability measures, which is a fundamental concept in many fields of mathematics. It has also been used in the development of new mathematical theories and applications, making it an important tool for researchers and practitioners alike.

• Topology and Analysis
Replies
6
Views
700
• Calculus
Replies
11
Views
786
• Topology and Analysis
Replies
5
Views
1K
• Topology and Analysis
Replies
2
Views
3K
• Differential Geometry
Replies
4
Views
3K
• Topology and Analysis
Replies
2
Views
526
• Topology and Analysis
Replies
5
Views
1K
• Optics
Replies
6
Views
1K
• Topology and Analysis
Replies
6
Views
1K
• Topology and Analysis
Replies
1
Views
989