# 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.

#### References

 [1] H. Minkowski, "Volumen und Oberfläche" Math. Ann. , 57 (1903) pp. 447–495 [2] A.V. Pogorelov, "The Minkowski multidimensional problem" , Winston (1978) (Translated from Russian) [3] H. Busemann, "Convex surfaces" , Interscience (1958)