Minkowski problem

In differential geometry, the Minkowski problem, named after Hermann Minkowski, asks, for a given strictly positive real function ƒ defined on sphere, for a strictly convex compact surface S whose Gaussian curvature at the point x equals ƒ(n(x)) , where n(x) denotes the normal to S at x. Eugenio Calabi stated: "From the geometric view point the Minkowski problem is the Rosetta Stone, from which several related problems can be solved."^{[citation needed]}

The problem of radiolocation is easily reduced to the Minkowski problem in Euclidean 3-space: restoration of convex shape over the given Gauss surface curvature. The inverse problem of the short-wave diffraction is reduced to the Minkowski problem. The Minkowski problem is the basis of the mathematical theory of diffraction as well as for the physical theory of diffraction. In the 1960s Petr Ufimtsev (P. Ya. Ufimtsev) began developing a high-frequency asymptotic theory for predicting the scattering of electromagnetic waves from two-dimensional and three-dimensional objects. Now this theory is well known as the physical theory of diffraction (PTD). This theory played the main role in the design of American stealth-aircraft F-117 and B-2.

In 1953 Louis Nirenberg published the solutions of two long standing open problems, the Weyl problem and the Minkowski problem in Euclidean 3-space. L. Nirenberg's solution of the Minkowski problem was a milestone in global geometry.

A. V. Pogorelov received Ukraine State Prize (1973) for resolving the multidimensional Minkowski problem in Euclidean spaces. Pogorelov resolved the Weyl problem in Riemannian space in 1969.

Shing-Tung Yau's joint work with S. Y. Cheng gives a complete proof of the higher dimensional Minkowski problem in Euclidean spaces. Shing-Tung Yau received the Fields Medal at the International Congress of Mathematicians in Warsaw in 1982 for his work in global differential geometry and elliptic partial differential equations, particularly for solving such difficult problems as the Calabi conjecture of 1954, and a problem of Hermann Minkowski in Euclidean spaces concerning the Dirichlet problem for the real Monge–Ampère equation.

References

• Minkowski, H. (1903). "Volumen und Oberfläche". Mathematische Annalen 57 (4): 447–495. doi:10.1007/BF01445180. 
• Cheng, Shiu Yuen; Yau, Shing Tung (1976). "On the regularity of the solution of the n-dimensional Minkowski problem". Comm. Pure Appl. Math. 29 (5): 495–516. doi:10.1002/cpa.3160290504. 
• Nirenberg, L. (1953). "The Weyl and Minkowski problems in differential geometry in the large". Comm. Pure App. Math. 6 (3): 337–394. doi:10.1002/cpa.3160060303. 
• Pogorelov, A. V. (1979) The Minkowsky multidimensional problem. Washington: Scripta, 97 p.
• Thompson, Donald O. Dale E. Chimenti. Review of Progress in Quantitative Nondestructive Evaluation.
• Equations of Mathematical Diffraction Theory, Mezhlum A. Sumbatyan, Rostov State University, Russia; Antonio Scalia, University of Catania, Italy.
• Ufimtsev, P. Ya. (2007). Fundamentals of the Physical Theory of Diffraction. Hoboken, New Jersey: Wiley & Sons. ISBN 047009771X. 
• Ufimtsev, P. Ya. (1962). Method of Edge Waves in the Physical Theory of Diffraction. Moscow: Soviet Radio.