Minkowski problem
Encyclopedia
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."
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.
Hermann Minkowski
Hermann Minkowski was a German mathematician of Ashkenazi Jewish descent, who created and developed the geometry of numbers and who used geometrical methods to solve difficult problems in number theory, mathematical physics, and the theory of relativity.- Life and work :Hermann Minkowski was born...
, asks, for a given strictly positive real function ƒ defined on sphere, for a strictly convex compact
Compact space
In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
surface
Surface
In mathematics, specifically in topology, a surface is a two-dimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space R3 — for example, the surface of a ball...
S whose Gaussian curvature
Gaussian curvature
In differential geometry, the Gaussian curvature or Gauss curvature of a point on a surface is the product of the principal curvatures, κ1 and κ2, of the given point. It is an intrinsic measure of curvature, i.e., its value depends only on how distances are measured on the surface, not on the way...
at the point x equals ƒ(n(x)) , where n(x) denotes the normal to S at x. Eugenio Calabi
Eugenio Calabi
Eugenio Calabi is a Italian American mathematician and professor emeritus at the University of Pennsylvania, specializing in differential geometry, partial differential equations and their applications....
stated: "From the geometric view point the Minkowski problem is the Rosetta Stone
Rosetta Stone
The Rosetta Stone is an ancient Egyptian granodiorite stele inscribed with a decree issued at Memphis in 196 BC on behalf of King Ptolemy V. The decree appears in three scripts: the upper text is Ancient Egyptian hieroglyphs, the middle portion Demotic script, and the lowest Ancient Greek...
, from which several related problems can be solved."
The problem of radiolocation
Radiolocation
Radiolocating is the process of finding the location of something through the use of radio waves. It generally refers to passive uses, particularly radar—as well as detecting buried cables, water mains, and other public utilities. It is similar to radionavigation, but radiolocation usually...
is easily reduced to the Minkowski problem in Euclidean 3-space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
: 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
Diffraction
Diffraction refers to various phenomena which occur when a wave encounters an obstacle. Italian scientist Francesco Maria Grimaldi coined the word "diffraction" and was the first to record accurate observations of the phenomenon in 1665...
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
Fields Medal
The Fields Medal, officially known as International Medal for Outstanding Discoveries in Mathematics, is a prize awarded to two, three, or four mathematicians not over 40 years of age at each International Congress of the International Mathematical Union , a meeting that takes place every four...
at the International Congress of Mathematicians
International Congress of Mathematicians
The International Congress of Mathematicians is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union ....
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
Calabi conjecture
In mathematics, the Calabi conjecture was a conjecture about the existence of good Riemannian metrics on complex manifolds, made by and proved by ....
of 1954, and a problem of Hermann Minkowski
Hermann Minkowski
Hermann Minkowski was a German mathematician of Ashkenazi Jewish descent, who created and developed the geometry of numbers and who used geometrical methods to solve difficult problems in number theory, mathematical physics, and the theory of relativity.- Life and work :Hermann Minkowski was born...
in Euclidean spaces concerning the Dirichlet problem
Dirichlet problem
In mathematics, a Dirichlet problem is the problem of finding a function which solves a specified partial differential equation in the interior of a given region that takes prescribed values on the boundary of the region....
for the real Monge–Ampère equation.