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Does there exist a closed convex hyperplane  $  F $
for which the [[Gaussian curvature|Gaussian curvature]]  $  K ( \xi ) $
is a given function of the unit outward normal  $  \xi $? 
This problem was posed by H. Minkowski [[#References|[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 [[#References|[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 [[#References|[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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Minkowski,   "Volumen und Oberfläche"  ''Math. Ann.'' , '''57'''  (1903)  pp. 447–495</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.V. Pogorelov,   "The Minkowski multidimensional problem" , Winston  (1978)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  H. Busemann,   "Convex surfaces" , Interscience  (1958)</TD></TR></table>

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.V. Pogorelov,   "Extrinsic geometry of convex surfaces" , Amer. Math. Soc.  (1973)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M. Berger,   B. Gostiaux,   "Differential geometry: manifolds, curves, and surfaces" , Springer  (1988)  (Translated from French)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  R. Schneider,   "Boundary structure and curvature of convex bodies"  J. Tölke (ed.)  J.M. Wills (ed.) , ''Contributions to geometry'' , Birkhäuser  (1979)  pp. 13–59</TD></TR></table>