Does there exist a closed convex hyperplane for which the Gaussian curvature is a given function of the unit outward normal ? This problem was posed by H. Minkowski , to whom is due a generalized solution of the problem in the sense that it contains no information on the nature of regularity of , even if is an analytic function. He proved that if a continuous positive function , given on the hypersphere , satisfies the condition
then there exists a closed convex surface , which is moreover unique (up to a parallel translation), for which is the Gaussian curvature at a point with outward normal .
A regular solution of Minkowski's problem has been given by A.V. Pogorelov in 1971 (see ); he also considered certain questions in geometry and in the theory of differential equations bordering on this problem. Namely, he proved that if is of class , , then the surface is of class , , and if is analytic, then also turns out to be analytic.
A natural generalization of Minkowski's problem is the solution of the question of the existence of convex hypersurfaces with given elementary symmetric principal curvature functions of any given order , . In particular, for this is Christoffel's problem on the recovery of a surface from its mean curvature. A necessary condition for the solvability of this generalized Minkowski problem, analogous to (1), has the form
However, this condition is not sufficient (A.D. Aleksandrov, 1938, see ). There are examples of sufficient conditions:
where , , . Here the regularity of is as in the Minkowski problem. Using approximations these results turn out to be valid even for functions which are non-negative, symmetric and concave.
|||H. Minkowski, "Volumen und Oberfläche" Math. Ann. , 57 (1903) pp. 447–495|
|||A.V. Pogorelov, "The Minkowski multidimensional problem" , Winston (1978) (Translated from Russian)|
|||H. Busemann, "Convex surfaces" , Interscience (1958)|
|[a1]||A.V. Pogorelov, "Extrinsic geometry of convex surfaces" , Amer. Math. Soc. (1973) (Translated from Russian)|
|[a2]||M. Berger, B. Gostiaux, "Differential geometry: manifolds, curves, and surfaces" , Springer (1988) (Translated from French)|
|[a3]||R. Schneider, "Boundary structure and curvature of convex bodies" J. Tölke (ed.) J.M. Wills (ed.) , Contributions to geometry , Birkhäuser (1979) pp. 13–59|
Minkowski problem. M.I. Voitsekhovskii (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Minkowski_problem&oldid=16770