differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, lin ...

, the Minkowski problem, named after

Hermann Minkowski Hermann Minkowski (22 June 1864 – 12 January 1909) was a mathematician and professor at the University of Königsberg, the University of Zürich, and the University of Göttingen, described variously as German, Polish, Lithuanian-German, o ...

, asks for the construction of a strictly convex

compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact, a type of agreement used by U.S. states * Blood compact, an ancient ritual of the Philippines * Compact government, a t ...

surface A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...

''S'' whose

Gaussian curvature In differential geometry, the Gaussian curvature or Gauss curvature of a smooth Surface (topology), surface in three-dimensional space at a point is the product of the principal curvatures, and , at the given point: K = \kappa_1 \kappa_2. For ...

is specified. More precisely, the input to the problem is a strictly positive real function ''ƒ'' defined on a sphere, and the surface that is to be constructed should have

''ƒ''(''n''(''x'')) at the point ''x'', where ''n''(''x'') denotes the normal to ''S'' at ''x''.

Eugenio Calabi Eugenio Calabi (May 11, 1923 – September 25, 2023) was an Italian-born American mathematician and the Thomas A. Scott Professor of Mathematics at the University of Pennsylvania, specializing in differential geometry, partial differential equa ...

stated: "From the geometric view point it he Minkowski problemis the

Rosetta Stone The Rosetta Stone is a stele of granodiorite inscribed with three versions of a Rosetta Stone decree, decree issued in 196 BC during the Ptolemaic dynasty of ancient Egypt, Egypt, on behalf of King Ptolemy V Epiphanes. The top and middle texts ...

, from which several related problems can be solved." In full generality, the Minkowski problem asks for necessary and sufficient conditions on a non-negative Borel measure on the unit sphere ''S''^n-1 to be the surface area measure of a

convex body In mathematics, a convex body in n-dimensional Euclidean space \R^n is a compact convex set with non- empty interior. Some authors do not require a non-empty interior, merely that the set is non-empty. A convex body K is called symmetric if it ...

\mathbb^n

. Here the surface area measure ''S_K'' of a convex body ''K'' is the pushforward of the ''(n-1)''-dimensional Hausdorff measure restricted to the boundary of ''K'' via the

Gauss map In differential geometry, the Gauss map of a surface is a function that maps each point in the surface to its normal direction, a unit vector that is orthogonal to the surface at that point. Namely, given a surface ''X'' in Euclidean space R3 ...

. The Minkowski problem was solved by

Aleksandr Danilovich Aleksandrov Aleksandr Danilovich Aleksandrov (; 4 August 1912 – 27 July 1999) was a Soviet and Russian mathematician, physicist, philosopher and mountaineer. Personal life Aleksandr Aleksandrov was born in 1912 in Volyn, Ryazan Oblast. His father was ...

Werner Fenchel Moritz Werner Fenchel (; 3 May 1905 – 24 January 1988) was a German-Danish mathematician known for his contributions to geometry and to optimization theory. Fenchel established the basic results of convex analysis and nonlinear opti ...

and

Børge Jessen Børge Christian Jessen (19 June 1907 – 20 March 1993) was a Danish mathematician best known for his work in analysis, specifically on the Riemann zeta function, and in geometry, specifically on Hilbert's third problem. Early years Jessen was ...

: a Borel measure ''μ'' on the unit sphere is the surface area measure of a convex body if and only if ''μ'' has centroid at the origin and is not concentrated on a great subsphere. The convex body is then uniquely determined by ''μ'' up to translations. The Minkowski problem, despite its clear geometric origin, is found to have its appearance in many places. The problem of

radiolocation Radiolocation, also known as radiolocating or radiopositioning, is the process of finding the location of something through the use of radio waves. It generally refers to passive, particularly radar—as well as detecting buried cables, wate ...

is easily reduced to the Minkowski problem in

Euclidean 3-space In geometry, a three-dimensional space (3D space, 3-space or, rarely, tri-dimensional space) is a mathematical space in which three values (''coordinates'') are required to determine the position of a point. Most commonly, it is the three-dim ...

: 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 is the deviation of waves from straight-line propagation without any change in their energy due to an obstacle or through an aperture. The diffracting object or aperture effectively becomes a secondary source of the Wave propagation ...

as well as for the physical theory of diffraction. In 1953

Louis Nirenberg Louis Nirenberg (February 28, 1925 – January 26, 2020) was a Canadian-American mathematician, considered one of the most outstanding Mathematical analysis, mathematicians of the 20th century. Nearly all of his work was in the field of par ...

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. He has been selected to be the first recipient of the Chern Medal (in 2010) for his role in the formulation of the modern theory of non-linear elliptic partial differential equations, particularly for solving the Weyl problem and the Minkowski problems in Euclidean 3-space. 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 Shing-Tung Yau (; ; born April 4, 1949) is a Chinese-American mathematician. He is the director of the Yau Mathematical Sciences Center at Tsinghua University and professor emeritus at Harvard University. Until 2022, Yau was the William Caspar ...

's joint work with

Shiu-Yuen Cheng Shiu-Yuen Cheng (鄭紹遠) is a Hong Kong mathematician. He is currently the Chair Professor of Mathematics at the Hong Kong University of Science and Technology. Cheng received his Ph.D. in 1974, under the supervision of Shiing-Shen Chern, from ...

gives a complete proof of the higher-dimensional Minkowski problem in Euclidean spaces. Shing-Tung Yau received the

Fields Medal The Fields Medal is a prize awarded to two, three, or four mathematicians under 40 years of age at the International Congress of Mathematicians, International Congress of the International Mathematical Union (IMU), a meeting that takes place e ...

at the

International Congress of Mathematicians The International Congress of Mathematicians (ICM) is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union (IMU). The Fields Medals, the IMU Abacus Medal (known before ...

in Warsaw in 1982 for his work in global differential geometry and

elliptic partial differential equation In mathematics, an elliptic partial differential equation is a type of partial differential equation (PDE). In mathematical modeling, elliptic PDEs are frequently used to model steady states, unlike parabolic PDE and hyperbolic PDE which gene ...

s, particularly for solving such difficult problems as the

Calabi conjecture In the mathematical field of differential geometry, the Calabi conjecture was a conjecture about the existence of certain kinds of Riemannian metrics on certain complex manifolds, made by . It was proved by , who received the Fields Medal and Oswa ...

of 1954, and a problem of

in Euclidean spaces concerning the

Dirichlet problem In mathematics, a Dirichlet problem asks for a function which solves a specified partial differential equation (PDE) in the interior of a given region that takes prescribed values on the boundary of the region. The Dirichlet problem can be solved ...

for the real

Monge–Ampère equation In mathematics, a (real) Monge–Ampère equation is a nonlinear second-order partial differential equation of special kind. A second-order equation for the unknown function ''u'' of two variables ''x'',''y'' is of Monge–Ampère type if it is l ...

References

Further reading