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Does there exist a closed convex hyperplane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064070/m0640701.png" /> for which the [[Gaussian curvature|Gaussian curvature]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064070/m0640702.png" /> is a given function of the unit outward normal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064070/m0640703.png" />? 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064070/m0640704.png" />, even if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064070/m0640705.png" /> is an analytic function. He proved that if a continuous positive function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064070/m0640706.png" />, given on the hypersphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064070/m0640707.png" />, satisfies the condition
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064070/m0640708.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
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then there exists a closed convex surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064070/m0640709.png" />, which is moreover unique (up to a parallel translation), for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064070/m06407010.png" /> is the Gaussian curvature at a point with outward normal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064070/m06407011.png" />.
+
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
  
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064070/m06407012.png" /> is of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064070/m06407013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064070/m06407014.png" />, then the surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064070/m06407015.png" /> is of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064070/m06407016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064070/m06407017.png" />, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064070/m06407018.png" /> is analytic, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064070/m06407019.png" /> also turns out to be analytic.
+
$$ \tag{1 }
 +
\int\limits _ { S } \xi
 +
\frac{d s }{K ( \xi ) }
 +
  = 0 ,
 +
$$
  
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064070/m06407020.png" /> of any given order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064070/m06407021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064070/m06407022.png" />. In particular, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064070/m06407023.png" /> 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
+
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 $.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064070/m06407024.png" /></td> </tr></table>
+
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:
 
However, this condition is not sufficient (A.D. Aleksandrov, 1938, see [[#References|[3]]]). There are examples of sufficient conditions:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064070/m06407025.png" /></td> </tr></table>
+
$$
 +
\int\limits _ { S } \xi \Phi _  \nu  ( \xi )  d S  = 0 ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064070/m06407026.png" /></td> </tr></table>
+
$$
 +
\left ( 1 -  
 +
\frac{1}{n}
 +
\right ) ^ {1 / 2 ( \nu - 1 ) }
 +
\max ( \Phi _ {\nu , t }  - \Phi _ {\nu , t }  ^ {\prime\prime} )  < \Phi _ {\nu , t }  ( \xi ) ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064070/m06407027.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064070/m06407028.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064070/m06407029.png" />. Here the regularity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064070/m06407030.png" /> is as in the Minkowski problem. Using approximations these results turn out to be valid even for functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064070/m06407031.png" /> which are non-negative, symmetric and concave.
+
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====
 
====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>
 
<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====
 
====Comments====
 
  
 
====References====
 
====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>
 
<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>

Latest revision as of 08:00, 6 June 2020


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)

Comments

References

[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
How to Cite This Entry:
Minkowski problem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Minkowski_problem&oldid=47854
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article