# What is Minkowski problem?

I have a question about 'Minkowski problem' related to Gaussian curvature. I searched 'Minkowski problem' in Google, Almost all of them were related to Einstein's theory of relativity. So I ask about this here!

As far as I know, the Minkowski problem is:

The classical Minkowski problem can be stated as follows: Given a smooth closed curve in the plane, can we find a convex, closed curve with the same length and the same Gaussian curvature at each point?

At the time Minkowski posed this problem, there was a growing interest in the theory of curves and surfaces. Mathematicians were exploring various geometric properties of curves and surfaces, including curvature. The concept of Gaussian curvature, which measures the curvature of a surface at each point, had been developed by Carl Friedrich Gauss in the early 19th century.

The difficulty that mathematicians faced when the classical Minkowski problem arose was finding a convex curve with the same length and the same Gaussian curvature.

But I'm not sure if this is correct, and even if this is correct, I want to know more about Minkowski problem's historical background and motivation, so I'm posting this question.

• Have you looked at the book Convex Bodies: The Brunn−Minkowski Theory by Rolf Schneider (2013, 2nd edition)? I don't know anything about this topic (nor do I have access to this book), but 2-3 minutes of googling led me to several places in which this book was recommended for historical information (for example, "(or see [9] for history)" in this paper). Commented Jul 4, 2023 at 14:14
• Thank you for your response! I'll find it Commented Jul 4, 2023 at 21:52
• Do you have a references? You must have read about it somewhere Commented Jul 7, 2023 at 18:15
• To be honest, I used chat gpt. Commented Jul 8, 2023 at 22:09

## 1 Answer

There are several problems which can be called "Minkowski problem". Perhaps the most famous one is this problem from differential geometry: Minkowski Problem.