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A branch of the theory of convex bodies concerned with the functionals that arise in the study of linear combinations of bodies (see [[Addition of sets|Addition of sets]]).
 
A branch of the theory of convex bodies concerned with the functionals that arise in the study of linear combinations of bodies (see [[Addition of sets|Addition of sets]]).
  
The volume <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m0642601.png" /> of a linear combination <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m0642602.png" /> of convex bodies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m0642603.png" /> in a Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m0642604.png" /> with coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m0642605.png" /> is a homogeneous polynomial of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m0642606.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m0642607.png" />:
+
The volume $  V $
 +
of a linear combination $  \sum _ {i= 1}  ^ {r} \lambda _ {i} K _ {i} $
 +
of convex bodies $  K _ {i} $
 +
in a Euclidean space $  \mathbf R  ^ {n} $
 +
with coefficients $  \lambda _ {i} \geq  0 $
 +
is a homogeneous polynomial of degree $  n $
 +
in $  \lambda _ {1} \dots \lambda _ {r} $:
  
<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/m064260/m0642608.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
+
$$ \tag{* }
 +
V \left ( \sum _ { i= 1 }^ { r }  \lambda _ {i} K _ {i} \right )  = \
 +
\sum _ {i _ {1} = 1 } ^ { r }  \dots \sum _ {i _ {n} = 1 } ^ { r }
 +
V _ {i _ {1}  \dots i _ {n} } \lambda _ {i _ {1}  \dots i _ {n} } .
 +
$$
  
The coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m0642609.png" /> are assumed to be symmetric with respect to permutations of the subscripts and are denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426010.png" />, since they depend only on the bodies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426011.png" />. These coefficients are called the mixed volumes of the bodies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426012.png" />.
+
The coefficients $  V _ {i _ {1}  \dots i _ {n} } $
 +
are assumed to be symmetric with respect to permutations of the subscripts and are denoted by $  V ( K _ {i _ {1}  } \dots K _ {i _ {n}  } ) $,  
 +
since they depend only on the bodies $  K _ {i _ {1}  } \dots K _ {i _ {n}  } $.  
 +
These coefficients are called the mixed volumes of the bodies $  K _ {i _ {1}  } \dots K _ {i _ {n}  } $.
  
The significance of this theory lies in the universality of the concept of mixed volumes: by substituting concrete bodies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426013.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426014.png" />, one can obtain various quantities related to a body <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426015.png" />. These include: its volume, its surface area, the surface integral of the elementary symmetric function of its principal curvatures (in the case of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426016.png" />-smooth body), and also the corresponding characteristics of its projections to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426017.png" />-dimensional planes, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426018.png" />. A special case of the expression (*) is the Steiner formula for volumes of parallel bodies in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426019.png" />:
+
The significance of this theory lies in the universality of the concept of mixed volumes: by substituting concrete bodies $  K _ {1} \dots K _ {n- 1} $
 +
into $  V ( K , K _ {1} \dots K _ {n- 1} ) $,  
 +
one can obtain various quantities related to a body $  K $.  
 +
These include: its volume, its surface area, the surface integral of the [[elementary symmetric function]] of its principal curvatures (in the case of a   $C^{2}$-smooth body), and also the corresponding characteristics of its projections to $  i $-
 +
dimensional planes, $  0 < i < n $.  
 +
A special case of the expression (*) is the Steiner formula for volumes of parallel bodies in $  \mathbf R  ^ {3} $:
  
<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/m064260/m06426020.png" /></td> </tr></table>
+
$$
 +
V _  \epsilon  = V + S \epsilon + \pi B \epsilon  ^ {2} +
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426021.png" /> is the volume, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426022.png" /> the surface area, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426023.png" /> the total mean curvature of the original body, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426024.png" /> the volume of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426025.png" />-neighbourhood of it. The mixed volume <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426026.png" /> is invariant under parallel translation of any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426027.png" />, is monotonic (with respect to inclusion of bodies), continuous and non-negative; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426028.png" /> if and only if it is possible to choose a segment in every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426029.png" /> such that these segments are linearly independent (see [[#References|[1]]]).
+
\frac{4}{3}
 +
\pi \epsilon  ^ {3} ,
 +
$$
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426030.png" /> is the projection of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426031.png" /> to a hypersurface orthogonal to a segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426032.png" /> of length one, then
+
where  $  V $
 +
is the volume,  $  S $
 +
the surface area,  $  B $
 +
the total mean curvature of the original body, and  $  V _  \epsilon  $
 +
the volume of an  $  \epsilon $-
 +
neighbourhood of it. The mixed volume  $  V ( K _ {1} \dots K _ {n} ) $
 +
is invariant under parallel translation of any  $  K _ {i} $,
 +
is monotonic (with respect to inclusion of bodies), continuous and non-negative;  $  V ( K _ {1} \dots K _ {n} ) > 0 $
 +
if and only if it is possible to choose a segment in every  $  K _ {i} $
 +
such that these segments are linearly independent (see [[#References|[1]]]).
  
<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/m064260/m06426033.png" /></td> </tr></table>
+
If  $  K  ^  \prime  $
 +
is the projection of  $  K $
 +
to a hypersurface orthogonal to a segment  $  e $
 +
of length one, then
  
The volume of the projection of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426034.png" /> to a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426035.png" />-dimensional subspace is called its <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426037.png" />-th cross-sectional measure (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426039.png" />-quermass). Establishing relations between the mean values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426040.png" /> of these measures is one of the concerns of [[Integral geometry|integral geometry]]. Up to a multiplicative constant, the functionals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426041.png" /> coincide with the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426043.png" />-th integral curvatures:
+
$$
 +
V ( K _ {1} \dots K _ {n- 1} , e )  = \
 +
n V ( K _ {1}  ^  \prime  \dots K _ {n- 1}  ^  \prime  ) .
 +
$$
  
<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/m064260/m06426044.png" /></td> </tr></table>
+
The volume of the projection of $K$
 +
to a $p$-dimensional subspace is called its $p$-th cross-sectional measure (or  $  p $-
 +
quermass). Establishing relations between the mean values  $  W _ {p} ( K) $
 +
of these measures is one of the concerns of [[Integral geometry|integral geometry]]. Up to a multiplicative constant, the functionals  $  W _ {p} ( K) $
 +
coincide with the  $  p $-th integral curvatures:
  
(<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426045.png" /> occurrences of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426046.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426047.png" /> occurrences of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426048.png" />), where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426049.png" /> is the unit sphere. For a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426050.png" />-smooth strictly convex body <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426051.png" />, the mixed volume <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426052.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426053.png" />, is equal to the integral of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426054.png" />-th elementary symmetric function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426055.png" /> of the principal radii of curvature, regarded as a function of the normal to the sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426056.png" />. In the case of a general convex body, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426057.png" /> is the total value of the measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426058.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426059.png" />, defined below and called the curvature function. (In the smooth case, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426060.png" /> is the density of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426061.png" />.) Just as the volume of the body <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426062.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426063.png" />-th of the integral of its [[Support function|support function]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426064.png" /> with respect to its surface function, i.e. the surface area of the image on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426065.png" /> under the spherical mapping (cf. [[Spherical map|Spherical map]]), the mixed volume of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426066.png" /> bodies can be written as the integral of the support function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426067.png" /> of one of them with respect to some measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426068.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426069.png" /> that depends on the other bodies, called the mixed surface function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426070.png" />:
+
$$
 +
V _ {p} ( K) = V ( {K \dots K } , {U \dots U } )
 +
$$
  
<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/m064260/m06426071.png" /></td> </tr></table>
+
( $  p $
 +
occurrences of  $  K $,
 +
$  n- p $
 +
occurrences of  $  U $),
 +
where  $  U $
 +
is the unit sphere. For a  $  C  ^ {2} $-
 +
smooth strictly convex body  $  K $,
 +
the mixed volume  $  V _ {p} ( K) $,
 +
$  0 < p < n $,
 +
is equal to the integral of the  $  p $-
 +
th elementary symmetric function  $  D _ {p} $
 +
of the principal radii of curvature, regarded as a function of the normal to the sphere  $  S  ^ {n-} 1 $.
 +
In the case of a general convex body,  $  V _ {p} ( K) $
 +
is the total value of the measure  $  \mu _ {p} $
 +
on  $  S  ^ {n-} 1 $,
 +
defined below and called the curvature function. (In the smooth case,  $  D _ {p} $
 +
is the density of  $  \mu _ {p} $.)
 +
Just as the volume of the body  $  K $
 +
is  $  1 / n $-
 +
th of the integral of its [[Support function|support function]]  $  K ( u ) $
 +
with respect to its surface function, i.e. the surface area of the image on  $  S  ^ {n-} 1 $
 +
under the spherical mapping (cf. [[Spherical map|Spherical map]]), the mixed volume of  $  n $
 +
bodies can be written as the integral of the support function  $  K _ {1} ( u) $
 +
of one of them with respect to some measure  $  \mu ( \omega ) = \mu ( K _ {2} \dots K _ {n} , \omega ) $
 +
on  $  S  ^ {n-} 1 $
 +
that depends on the other bodies, called the mixed surface function of  $  K _ {2} \dots K _ {n} $:
  
The curvature function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426072.png" /> is defined by the equation
+
$$
 +
V ( K _ {1} \dots K _ {n} )  =
 +
\frac{1}{n}
  
<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/m064260/m06426073.png" /></td> </tr></table>
+
\int\limits _ {S  ^ {n- 1} } K _ {1} ( u)  d \mu .
 +
$$
  
(<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426074.png" /> occurrences of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426075.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426076.png" /> occurrences of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426077.png" />).
+
The curvature function  $  \mu _ {p} ( \omega ) $
 +
is defined by the equation
 +
 
 +
$$
 +
\mu _ {p} ( \omega )  = \mu ( {K \dots K } ,\
 +
{U \dots U , \omega } )
 +
$$
 +
 
 +
( $  p $
 +
occurrences of $  K $,  
 +
$  n- p- 2 $
 +
occurrences of $  U $).
  
 
The main content of the theory of mixed volumes is formed by inequalities between mixed volumes (see , [[#References|[3]]]). They include the Minkowski inequality
 
The main content of the theory of mixed volumes is formed by inequalities between mixed volumes (see , [[#References|[3]]]). They include the Minkowski inequality
  
<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/m064260/m06426078.png" /></td> </tr></table>
+
$$
 +
V  ^ {n} ( K , L \dots L )  \geq  V ( K ) V  ^ {n- 1} ( L)
 +
$$
  
 
and the quadratic Minkowski inequality
 
and the quadratic Minkowski inequality
  
<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/m064260/m06426079.png" /></td> </tr></table>
+
$$
 +
V  ^ {2} ( K , L \dots L )  \geq  V ( L) V ( K , K , L \dots L ) .
 +
$$
  
 
Both of these are closely connected with the [[Brunn–Minkowski theorem|Brunn–Minkowski theorem]], which is true not only for convex bodies. The Aleksandrov–Fenchel inequality generalizes these, and has the following modification (see ):
 
Both of these are closely connected with the [[Brunn–Minkowski theorem|Brunn–Minkowski theorem]], which is true not only for convex bodies. The Aleksandrov–Fenchel inequality generalizes these, and has the following modification (see ):
  
<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/m064260/m06426080.png" /></td> </tr></table>
+
$$
 +
V  ^ {m} ( K _ {1} \dots K _ {m} , L _ {1} \dots L _ {n- m} ) \geq
 +
$$
  
<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/m064260/m06426081.png" /></td> </tr></table>
+
$$
 +
\geq  \
 +
\prod _ { i= } 1 ^ { m }  V ( K _ {i} \dots K _ {i} , L _ {1} \dots L _ {n- m} ) .
 +
$$
  
 
In particular,
 
In particular,
  
<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/m064260/m06426082.png" /></td> </tr></table>
+
$$
 
+
V ^ {n} ( K _ {1} \dots K _ {n} \geq \
A complete system of inequalities characterizing the mixed volume <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426083.png" /> has been obtained in the case of two bodies, and more general inequalities have been established (see [[#References|[4]]]).
+
V ( K _ {1} ) \dots V ( K _ {n} ) .
 
+
$$
Many geometric inequalities, such as the classical isoperimetric inequality (cf. [[Isoperimetric inequality, classical|Isoperimetric inequality, classical]]) and several of its refinements, are special cases of inequalities for mixed volumes of convex bodies. The extremum of one of the functionals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426084.png" /> when some other such functional is fixed is attained for a sphere. Inequalities in the theory of mixed volumes are used in proving the uniqueness of the solution of the generalized [[Minkowski problem|Minkowski problem]] (see ), in establishing stability in Minkowski problems (see [[#References|[5]]]) and Weil problems (see [[#References|[6]]]), and in the solution of the van der Waerden problem on permanents (see [[#References|[7]]]). Infinite-dimensional analogues of the concepts of the theory of mixed volumes have found application in the theory of Gaussian stochastic processes (see [[#References|[7]]]).
 
 
 
The theory of mixed volumes has deep connections with algebraic geometry. Given a polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426085.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426086.png" /> complex variables, one can define its Newton polyhedron <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426087.png" /> in the following way. To each monomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426088.png" /> occurring in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426089.png" /> with a non-zero coefficient, one assigns the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426090.png" />, and one defines <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426091.png" /> as the convex hull of these points. The typical number of solutions of the system of polynomial equations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426092.png" /> is equal to the mixed volume of the polyhedron <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426093.png" /> divided by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426094.png" />. Among other things, this allows one to give an algebraic proof of the Aleksandrov–Fenchel inequality (see [[#References|[10]]]).
 
 
 
In the theory of mixed volumes, a convex body is identified with its support function. This is extended to differences of these functions, and then to arbitrary continuous functions on a sphere (see , [[#References|[9]]]). Using the analogous decomposition of the vector of the centre of gravity of a body <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426095.png" /> multiplied by its volume, one can define the so-called mixed direction vectors, which are vector analogues of mixed volumes. The centres of gravity of a body <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426096.png" /> coincide up to a multiplicative constant with the mixed direction vectors of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426097.png" /> and the sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m064/m064260/m06426098.png" /> (see [[#References|[11]]]).
 
 
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Minkowski,  "Theorie der konvexen Körpern, insbesonder der Begründung ihres Oberflächenbegriffs" , ''Gesammelte Abhandlungen'' , '''2''' , Teubner (1911)  pp. 131–229</TD></TR><TR><TD valign="top">[2a]</TD> <TD valign="top">  A.D. Aleksandrov,  "Zur Theorie gemischter Volumina von konvexen Körpern I. Verallgemeinerungen einiger Begriffe der Theorie von konvexen Körpern"  ''Mat. Sb.'' , '''2''' :  5  (1937)  pp. 947–972</TD></TR><TR><TD valign="top">[2b]</TD> <TD valign="top">  A.D. Aleksandrov,  "Zur Theorie gemischter Volumina von konvexen Körpern II. Neue Ungleichungen zwischen den gemischten Volumina und ihre Anwendungen"  ''Mat. Sb.'' , '''2''' :  6  pp. 1205–1238</TD></TR><TR><TD valign="top">[2c]</TD> <TD valign="top">  A.D. Aleksandrov,  "Zur Theorie gemischter Volumina von konvexen Körpern III. Die Erweiterung zweier Lehrsätze Minkowski's über die konvexen Polyedern auf die beliebigen konvexen Körper"  ''Mat. Sb.'' , '''3''' :  1 (1938pp. 27–46</TD></TR><TR><TD valign="top">[2d]</TD> <TD valign="top">  A.D. Aleksandrov,  "Zur Theorie gemischter Volumina von konvexen Körpern IV. Die gemischten Diskriminanten und die gemischten Volumina"  ''Mat. Sb.'' , '''3''' :  2  pp. 227–251</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  H. Busemann,  "Convex surfaces" , Interscience  (1958)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  G. Shephard,  "Inequalities between mixed volumes of convex sets"  ''Mathematika'' , '''7''' :  14  (1960)  pp. 125–138</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  V.I. Diskant,  "Stability of solutions of Minkowski's equation"  ''Sib. Math. J.'' , '''14''' :  3  (1973)  pp. 466–469  ''Sibirsk. Mat. Zh.'' , '''14''' :  3  (1973)  pp. 669–673</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  Yu.A. Volkov,  "Estimate of deformation of a convex surface as a function of the change in internal metric"  ''Ukrain. Geom. Sb.'' , '''5–6'''  (1968)  pp. 44–69  (In Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  D.E. Knuth,  "A permanent inequality"  ''Amer. Math. Monthly'' , '''88'''  (1981)  pp. 731–740</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> V.N. Sudakov,  "Gaussian random processes and measures of solid angles in Hilbert space"  ''Soviet Math. Dokl.'' , '''12''' :  2  (1971)  pp. 412–415  ''Dokl. Akad. Nauk SSSR'' , '''197''' :  1 (1971) pp. 43–45</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  H. Busemann,  G. Ewald,  G. Shepard,  "Convex bodies and convexity on Grassmann cones I-IV"  ''Math. Ann.'' , '''151''' :  1  (1963)  pp. 1–41</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  A.G. Khovanskii,  "Algebra and mixed volumes"  Yu.D. Burago (ed.)  V.A. Zalgaller (ed.) , ''Geometric Inequalities'' , Springer  (1988) pp. 182–207  (Translated from Russian)</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top">  R. Schneider,  "Krümmungsschwerpunkte konvexer Körper I"  ''Abh. Math. Sem. Univ. Hamburg'' , '''37'''  (1972)  pp. 112–132</TD></TR></table>
 
  
 +
A complete system of inequalities characterizing the mixed volume  $  V ( K _ {1} \dots K _ {n} ) $
 +
has been obtained in the case of two bodies, and more general inequalities have been established (see [[#References|[4]]]).
  
 +
Many geometric inequalities, such as the classical isoperimetric inequality (cf. [[Isoperimetric inequality, classical|Isoperimetric inequality, classical]]) and several of its refinements, are special cases of inequalities for mixed volumes of convex bodies. The extremum of one of the functionals  $  V _ {p} ( K) $
 +
when some other such functional is fixed is attained for a sphere. Inequalities in the theory of mixed volumes are used in proving the uniqueness of the solution of the generalized [[Minkowski problem|Minkowski problem]] (see ), in establishing stability in Minkowski problems (see [[#References|[5]]]) and Weil problems (see [[#References|[6]]]), and in the solution of the van der Waerden problem on permanents (see [[#References|[7]]]). Infinite-dimensional analogues of the concepts of the theory of mixed volumes have found application in the theory of Gaussian stochastic processes (see [[#References|[7]]]).
  
====Comments====
+
The theory of mixed volumes has deep connections with algebraic geometry. Given a polynomial  $  f ( z _ {1} \dots z _ {n} ) $
 +
in  $  n $
 +
complex variables, one can define its Newton polyhedron  $  \mathop{\rm Nw} ( f  ) $
 +
in the following way. To each monomial  $  z _ {1} ^ {a _ {1} } \dots z _ {n} ^ {a _ {n} } $
 +
occurring in  $  f $
 +
with a non-zero coefficient, one assigns the point  $  ( a _ {1} \dots a _ {n} ) \in \mathbf R  ^ {n} $,
 +
and one defines  $  \mathop{\rm Nw} ( f  ) $
 +
as the convex hull of these points. The typical number of solutions of the system of polynomial equations  $  f _ {1} = \dots = f _ {n} = 0 $
 +
is equal to the mixed volume of the polyhedron  $  \mathop{\rm Nw} ( f _ {1} \dots f _ {n} ) $
 +
divided by  $  n ! $.
 +
Among other things, this allows one to give an algebraic proof of the Aleksandrov–Fenchel inequality (see [[#References|[10]]]).
  
 +
In the theory of mixed volumes, a convex body is identified with its support function. This is extended to differences of these functions, and then to arbitrary continuous functions on a sphere (see , [[#References|[9]]]). Using the analogous decomposition of the vector of the centre of gravity of a body  $  \sum _ {i=} 1  ^ {r} \lambda _ {i} K _ {i} $
 +
multiplied by its volume, one can define the so-called mixed direction vectors, which are vector analogues of mixed volumes. The centres of gravity of a body  $  K $
 +
coincide up to a multiplicative constant with the mixed direction vectors of  $  K $
 +
and the sphere  $  U $(
 +
see [[#References|[11]]]).
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R. Schneider,   "On the Aleksandrov–Fenchel inequality" ''Ann. New York Acad. Sci.'' , '''440''' (1985) pp. 132–141</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> R. Schneider,   "Valuations on convex bodies" P.M. Gruber (ed.) J.M. Wills (ed.) , ''Convexity and its applications'' , Birkhäuser (1983) pp. 170–247</TD></TR></table>
+
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> H. Minkowski, "Theorie der konvexen Körper, insbesondere Begründung ihres Oberflächenbegriffs" , ''Gesammelte Abhandlungen'' , '''2''' , Teubner (1911) pp. 131–229</TD></TR><TR><TD valign="top">[2a]</TD> <TD valign="top"> A.D. Aleksandrov, "Zur Theorie gemischter Volumina von konvexen Körpern I. Verallgemeinerungen einiger Begriffe der Theorie von konvexen Körpern" ''Mat. Sb.'' , '''2''' : 5 (1937) pp. 947–972</TD></TR><TR><TD valign="top">[2b]</TD> <TD valign="top"> A.D. Aleksandrov, "Zur Theorie gemischter Volumina von konvexen Körpern II. Neue Ungleichungen zwischen den gemischten Volumina und ihre Anwendungen" ''Mat. Sb.'' , '''2''' : 6 pp. 1205–1238</TD></TR><TR><TD valign="top">[2c]</TD> <TD valign="top"> A.D. Aleksandrov, "Zur Theorie gemischter Volumina von konvexen Körpern III. Die Erweiterung zweier Lehrsätze Minkowski's über die konvexen Polyeder auf die beliebigen konvexen Körper" ''Mat. Sb.'' , '''3''' : 1 (1938) pp. 27–46</TD></TR><TR><TD valign="top">[2d]</TD> <TD valign="top"> A.D. Aleksandrov, "Zur Theorie gemischter Volumina von konvexen Körpern IV. Die gemischten Diskriminanten und die gemischten Volumina" ''Mat. Sb.'' , '''3''' : 2 pp. 227–251</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> H. Busemann, "Convex surfaces" , Interscience (1958) {{MR|0105155}} {{ZBL|0196.55101}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> G. Shephard, "Inequalities between mixed volumes of convex sets" ''Mathematika'' , '''7''' : 14 (1960) pp. 125–138 {{MR|0146736}} {{ZBL|0108.35203}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> V.I. Diskant, "Stability of solutions of Minkowski's equation" ''Sib. Math. J.'' , '''14''' : 3 (1973) pp. 466–469 ''Sibirsk. Mat. Zh.'' , '''14''' : 3 (1973) pp. 669–673 {{MR|333988}} {{ZBL|}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> Yu.A. Volkov, "Estimate of deformation of a convex surface as a function of the change in internal metric" ''Ukrain. Geom. Sb.'' , '''5–6''' (1968) pp. 44–69 (In Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> D.E. Knuth, "A permanent inequality" ''Amer. Math. Monthly'' , '''88''' (1981) pp. 731–740 {{MR|0668399}} {{ZBL|0478.15004}} </TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> V.N. Sudakov, "Gaussian random processes and measures of solid angles in Hilbert space" ''Soviet Math. Dokl.'' , '''12''' : 2 (1971) pp. 412–415 ''Dokl. Akad. Nauk SSSR'' , '''197''' : 1 (1971) pp. 43–45 {{MR|0288832}} {{ZBL|0231.60025}} </TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top"> H. Busemann, G. Ewald, G. Shepard, "Convex bodies and convexity on Grassmann cones I-IV" ''Math. Ann.'' , '''151''' : 1 (1963) pp. 1–41 {{MR|0157286}} {{MR|0157287}} {{ZBL|0112.37301}} </TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top"> A.G. Khovanskii, "Algebra and mixed volumes" Yu.D. Burago (ed.) V.A. Zalgaller (ed.) , ''Geometric Inequalities'' , Springer (1988) pp. 182–207 (Translated from Russian)</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top"> R. Schneider, "Krümmungsschwerpunkte konvexer Körper I" ''Abh. Math. Sem. Univ. Hamburg'' , '''37''' (1972) pp. 112–132 {{MR|0307039}} {{ZBL|0229.52005}} </TD></TR>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top"> R. Schneider, "On the Aleksandrov–Fenchel inequality" ''Ann. New York Acad. Sci.'' , '''440''' (1985) pp. 132–141 {{MR|0809200}} {{ZBL|0567.52004}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> R. Schneider, "Valuations on convex bodies" P.M. Gruber (ed.) J.M. Wills (ed.) , ''Convexity and its applications'' , Birkhäuser (1983) pp. 170–247 {{MR|0731112}} {{ZBL|0534.52001}} </TD></TR></table>

Latest revision as of 05:48, 26 April 2023


A branch of the theory of convex bodies concerned with the functionals that arise in the study of linear combinations of bodies (see Addition of sets).

The volume $ V $ of a linear combination $ \sum _ {i= 1} ^ {r} \lambda _ {i} K _ {i} $ of convex bodies $ K _ {i} $ in a Euclidean space $ \mathbf R ^ {n} $ with coefficients $ \lambda _ {i} \geq 0 $ is a homogeneous polynomial of degree $ n $ in $ \lambda _ {1} \dots \lambda _ {r} $:

$$ \tag{* } V \left ( \sum _ { i= 1 }^ { r } \lambda _ {i} K _ {i} \right ) = \ \sum _ {i _ {1} = 1 } ^ { r } \dots \sum _ {i _ {n} = 1 } ^ { r } V _ {i _ {1} \dots i _ {n} } \lambda _ {i _ {1} \dots i _ {n} } . $$

The coefficients $ V _ {i _ {1} \dots i _ {n} } $ are assumed to be symmetric with respect to permutations of the subscripts and are denoted by $ V ( K _ {i _ {1} } \dots K _ {i _ {n} } ) $, since they depend only on the bodies $ K _ {i _ {1} } \dots K _ {i _ {n} } $. These coefficients are called the mixed volumes of the bodies $ K _ {i _ {1} } \dots K _ {i _ {n} } $.

The significance of this theory lies in the universality of the concept of mixed volumes: by substituting concrete bodies $ K _ {1} \dots K _ {n- 1} $ into $ V ( K , K _ {1} \dots K _ {n- 1} ) $, one can obtain various quantities related to a body $ K $. These include: its volume, its surface area, the surface integral of the elementary symmetric function of its principal curvatures (in the case of a $C^{2}$-smooth body), and also the corresponding characteristics of its projections to $ i $- dimensional planes, $ 0 < i < n $. A special case of the expression (*) is the Steiner formula for volumes of parallel bodies in $ \mathbf R ^ {3} $:

$$ V _ \epsilon = V + S \epsilon + \pi B \epsilon ^ {2} + \frac{4}{3} \pi \epsilon ^ {3} , $$

where $ V $ is the volume, $ S $ the surface area, $ B $ the total mean curvature of the original body, and $ V _ \epsilon $ the volume of an $ \epsilon $- neighbourhood of it. The mixed volume $ V ( K _ {1} \dots K _ {n} ) $ is invariant under parallel translation of any $ K _ {i} $, is monotonic (with respect to inclusion of bodies), continuous and non-negative; $ V ( K _ {1} \dots K _ {n} ) > 0 $ if and only if it is possible to choose a segment in every $ K _ {i} $ such that these segments are linearly independent (see [1]).

If $ K ^ \prime $ is the projection of $ K $ to a hypersurface orthogonal to a segment $ e $ of length one, then

$$ V ( K _ {1} \dots K _ {n- 1} , e ) = \ n V ( K _ {1} ^ \prime \dots K _ {n- 1} ^ \prime ) . $$

The volume of the projection of $K$ to a $p$-dimensional subspace is called its $p$-th cross-sectional measure (or $ p $- quermass). Establishing relations between the mean values $ W _ {p} ( K) $ of these measures is one of the concerns of integral geometry. Up to a multiplicative constant, the functionals $ W _ {p} ( K) $ coincide with the $ p $-th integral curvatures:

$$ V _ {p} ( K) = V ( {K \dots K } , {U \dots U } ) $$

( $ p $ occurrences of $ K $, $ n- p $ occurrences of $ U $), where $ U $ is the unit sphere. For a $ C ^ {2} $- smooth strictly convex body $ K $, the mixed volume $ V _ {p} ( K) $, $ 0 < p < n $, is equal to the integral of the $ p $- th elementary symmetric function $ D _ {p} $ of the principal radii of curvature, regarded as a function of the normal to the sphere $ S ^ {n-} 1 $. In the case of a general convex body, $ V _ {p} ( K) $ is the total value of the measure $ \mu _ {p} $ on $ S ^ {n-} 1 $, defined below and called the curvature function. (In the smooth case, $ D _ {p} $ is the density of $ \mu _ {p} $.) Just as the volume of the body $ K $ is $ 1 / n $- th of the integral of its support function $ K ( u ) $ with respect to its surface function, i.e. the surface area of the image on $ S ^ {n-} 1 $ under the spherical mapping (cf. Spherical map), the mixed volume of $ n $ bodies can be written as the integral of the support function $ K _ {1} ( u) $ of one of them with respect to some measure $ \mu ( \omega ) = \mu ( K _ {2} \dots K _ {n} , \omega ) $ on $ S ^ {n-} 1 $ that depends on the other bodies, called the mixed surface function of $ K _ {2} \dots K _ {n} $:

$$ V ( K _ {1} \dots K _ {n} ) = \frac{1}{n} \int\limits _ {S ^ {n- 1} } K _ {1} ( u) d \mu . $$

The curvature function $ \mu _ {p} ( \omega ) $ is defined by the equation

$$ \mu _ {p} ( \omega ) = \mu ( {K \dots K } ,\ {U \dots U , \omega } ) $$

( $ p $ occurrences of $ K $, $ n- p- 2 $ occurrences of $ U $).

The main content of the theory of mixed volumes is formed by inequalities between mixed volumes (see , [3]). They include the Minkowski inequality

$$ V ^ {n} ( K , L \dots L ) \geq V ( K ) V ^ {n- 1} ( L) $$

and the quadratic Minkowski inequality

$$ V ^ {2} ( K , L \dots L ) \geq V ( L) V ( K , K , L \dots L ) . $$

Both of these are closely connected with the Brunn–Minkowski theorem, which is true not only for convex bodies. The Aleksandrov–Fenchel inequality generalizes these, and has the following modification (see ):

$$ V ^ {m} ( K _ {1} \dots K _ {m} , L _ {1} \dots L _ {n- m} ) \geq $$

$$ \geq \ \prod _ { i= } 1 ^ { m } V ( K _ {i} \dots K _ {i} , L _ {1} \dots L _ {n- m} ) . $$

In particular,

$$ V ^ {n} ( K _ {1} \dots K _ {n} ) \geq \ V ( K _ {1} ) \dots V ( K _ {n} ) . $$

A complete system of inequalities characterizing the mixed volume $ V ( K _ {1} \dots K _ {n} ) $ has been obtained in the case of two bodies, and more general inequalities have been established (see [4]).

Many geometric inequalities, such as the classical isoperimetric inequality (cf. Isoperimetric inequality, classical) and several of its refinements, are special cases of inequalities for mixed volumes of convex bodies. The extremum of one of the functionals $ V _ {p} ( K) $ when some other such functional is fixed is attained for a sphere. Inequalities in the theory of mixed volumes are used in proving the uniqueness of the solution of the generalized Minkowski problem (see ), in establishing stability in Minkowski problems (see [5]) and Weil problems (see [6]), and in the solution of the van der Waerden problem on permanents (see [7]). Infinite-dimensional analogues of the concepts of the theory of mixed volumes have found application in the theory of Gaussian stochastic processes (see [7]).

The theory of mixed volumes has deep connections with algebraic geometry. Given a polynomial $ f ( z _ {1} \dots z _ {n} ) $ in $ n $ complex variables, one can define its Newton polyhedron $ \mathop{\rm Nw} ( f ) $ in the following way. To each monomial $ z _ {1} ^ {a _ {1} } \dots z _ {n} ^ {a _ {n} } $ occurring in $ f $ with a non-zero coefficient, one assigns the point $ ( a _ {1} \dots a _ {n} ) \in \mathbf R ^ {n} $, and one defines $ \mathop{\rm Nw} ( f ) $ as the convex hull of these points. The typical number of solutions of the system of polynomial equations $ f _ {1} = \dots = f _ {n} = 0 $ is equal to the mixed volume of the polyhedron $ \mathop{\rm Nw} ( f _ {1} \dots f _ {n} ) $ divided by $ n ! $. Among other things, this allows one to give an algebraic proof of the Aleksandrov–Fenchel inequality (see [10]).

In the theory of mixed volumes, a convex body is identified with its support function. This is extended to differences of these functions, and then to arbitrary continuous functions on a sphere (see , [9]). Using the analogous decomposition of the vector of the centre of gravity of a body $ \sum _ {i=} 1 ^ {r} \lambda _ {i} K _ {i} $ multiplied by its volume, one can define the so-called mixed direction vectors, which are vector analogues of mixed volumes. The centres of gravity of a body $ K $ coincide up to a multiplicative constant with the mixed direction vectors of $ K $ and the sphere $ U $( see [11]).

References

[1] H. Minkowski, "Theorie der konvexen Körper, insbesondere Begründung ihres Oberflächenbegriffs" , Gesammelte Abhandlungen , 2 , Teubner (1911) pp. 131–229
[2a] A.D. Aleksandrov, "Zur Theorie gemischter Volumina von konvexen Körpern I. Verallgemeinerungen einiger Begriffe der Theorie von konvexen Körpern" Mat. Sb. , 2 : 5 (1937) pp. 947–972
[2b] A.D. Aleksandrov, "Zur Theorie gemischter Volumina von konvexen Körpern II. Neue Ungleichungen zwischen den gemischten Volumina und ihre Anwendungen" Mat. Sb. , 2 : 6 pp. 1205–1238
[2c] A.D. Aleksandrov, "Zur Theorie gemischter Volumina von konvexen Körpern III. Die Erweiterung zweier Lehrsätze Minkowski's über die konvexen Polyeder auf die beliebigen konvexen Körper" Mat. Sb. , 3 : 1 (1938) pp. 27–46
[2d] A.D. Aleksandrov, "Zur Theorie gemischter Volumina von konvexen Körpern IV. Die gemischten Diskriminanten und die gemischten Volumina" Mat. Sb. , 3 : 2 pp. 227–251
[3] H. Busemann, "Convex surfaces" , Interscience (1958) MR0105155 Zbl 0196.55101
[4] G. Shephard, "Inequalities between mixed volumes of convex sets" Mathematika , 7 : 14 (1960) pp. 125–138 MR0146736 Zbl 0108.35203
[5] V.I. Diskant, "Stability of solutions of Minkowski's equation" Sib. Math. J. , 14 : 3 (1973) pp. 466–469 Sibirsk. Mat. Zh. , 14 : 3 (1973) pp. 669–673 MR333988
[6] Yu.A. Volkov, "Estimate of deformation of a convex surface as a function of the change in internal metric" Ukrain. Geom. Sb. , 5–6 (1968) pp. 44–69 (In Russian)
[7] D.E. Knuth, "A permanent inequality" Amer. Math. Monthly , 88 (1981) pp. 731–740 MR0668399 Zbl 0478.15004
[8] V.N. Sudakov, "Gaussian random processes and measures of solid angles in Hilbert space" Soviet Math. Dokl. , 12 : 2 (1971) pp. 412–415 Dokl. Akad. Nauk SSSR , 197 : 1 (1971) pp. 43–45 MR0288832 Zbl 0231.60025
[9] H. Busemann, G. Ewald, G. Shepard, "Convex bodies and convexity on Grassmann cones I-IV" Math. Ann. , 151 : 1 (1963) pp. 1–41 MR0157286 MR0157287 Zbl 0112.37301
[10] A.G. Khovanskii, "Algebra and mixed volumes" Yu.D. Burago (ed.) V.A. Zalgaller (ed.) , Geometric Inequalities , Springer (1988) pp. 182–207 (Translated from Russian)
[11] R. Schneider, "Krümmungsschwerpunkte konvexer Körper I" Abh. Math. Sem. Univ. Hamburg , 37 (1972) pp. 112–132 MR0307039 Zbl 0229.52005
[a1] R. Schneider, "On the Aleksandrov–Fenchel inequality" Ann. New York Acad. Sci. , 440 (1985) pp. 132–141 MR0809200 Zbl 0567.52004
[a2] R. Schneider, "Valuations on convex bodies" P.M. Gruber (ed.) J.M. Wills (ed.) , Convexity and its applications , Birkhäuser (1983) pp. 170–247 MR0731112 Zbl 0534.52001
How to Cite This Entry:
Mixed-volume theory. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Mixed-volume_theory&oldid=11286
This article was adapted from an original article by Yu.D. Burago (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article