Minkowski problem

Does there exist a closed convex hyperplane $ F $ for which the Gaussian curvature $ K ( \xi ) $ is a given function of the unit outward normal $ \xi $? This problem was posed by H. Minkowski [1], to whom is due a generalized solution of the problem in the sense that it contains no information on the nature of regularity of $ F $, even if $ K ( \xi ) $ is an analytic function. He proved that if a continuous positive function $ K ( \xi ) $, given on the hypersphere $ S $, satisfies the condition

$$ \tag{1 } \int\limits _ { S } \xi \frac{d s }{K ( \xi ) } = 0 , $$

then there exists a closed convex surface $ F $, which is moreover unique (up to a parallel translation), for which $ K ( \xi ) $ is the Gaussian curvature at a point with outward normal $ \xi $.

A regular solution of Minkowski's problem has been given by A.V. Pogorelov in 1971 (see [2]); he also considered certain questions in geometry and in the theory of differential equations bordering on this problem. Namely, he proved that if $ K ( \xi ) $ is of class $ C ^ {m} $, $ m \geq 3 $, then the surface $ F $ is of class $ C ^ {m + 1 , \alpha } $, $ \alpha > 0 $, and if $ K ( \xi ) $ is analytic, then $ F $ 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 $ \phi _ \nu ( \xi ) $ of any given order $ \nu $, $ \nu \leq n = \mathop{\rm dim} F $. In particular, for $ \nu = 1 $ 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

$$ \int\limits _ { S } \xi \phi _ \nu ( \xi ) d S = 0 . $$

However, this condition is not sufficient (A.D. Aleksandrov, 1938, see [3]). There are examples of sufficient conditions:

$$ \int\limits _ { S } \xi \Phi _ \nu ( \xi ) d S = 0 , $$

$$ \left ( 1 - \frac{1}{n} \right ) ^ {1 / 2 ( \nu - 1 ) } \max ( \Phi _ {\nu , t } - \Phi _ {\nu , t } ^ {\prime\prime} ) < \Phi _ {\nu , t } ( \xi ) , $$

where $ \Phi _ {\nu , t } = ( \phi _ {\nu , t } / C _ {n} ^ \nu ) ^ {1/n} $, $ \phi _ {\nu , t } = t \phi _ \nu + 1 - t $, $ 0 \leq t \leq 1 $. Here the regularity of $ F $ is as in the Minkowski problem. Using approximations these results turn out to be valid even for functions $ \phi ( \xi ) $ which are non-negative, symmetric and concave.

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