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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]]).

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 [[#References|[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|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|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} $:

$$ 
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 , [[#References|[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|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 [[#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]]]).

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====
<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>