The origins of the theory of Mixed Volumes go back to Minkowski, who really created it. For a presentation of the theory in its original form and its history we can consult the book [160] of Bonnesen and Fenchel. With the contributions of Aleksandrov [14] and, independently, Fenchel-Jessen [348] the theory got a more elegant and general form, namely also with the introduction of the concept of mixed area measure.

Once the notion of mixed volume has been developed, the formula (12.1) can serve as the starting point for the definition of mixed area measure.

Given convex bodies, \(\Phi _1, \ldots , \Phi _{n-1}\), a convex body \(\Phi \) with support function \(P_{\Phi }: {\mathbb {S}}^{n-1} \rightarrow {\mathbb {E}}\) defines a linear functional \(\mathcal{F}_{\Phi _1, \ldots , \Phi _{n-1}}\) on the subset of the normed real vector space \(\mathcal{C}({\mathbb {S}}^{n-1})\) consisting of the support functions of all convex bodies \(\Phi \), via \(\mathcal{{F}}_{\Phi _1 , \ldots , \Phi _{n-1}} (\mathcal{{P}}_{\Phi }) := \mathbf {V}( \Phi , \Phi _1, \ldots , \Phi _{n-1})\). It can be shown that \(\mathcal{{F}}_{\Phi _1 , \ldots , \Phi _{n-1}}\) has a unique extension to a continuous linear functional on \(\mathcal {C}(\mathbb {S}^{n-1})\). By the Riesz Representation Theorem there exists a unique measure \(\mathbf {S}( \Phi _1,\dots ,\Phi _{n-1}, *)\) for which (12.1) holds true. The properties of the mixed area measure \(\mathbf {S}\) can be deduced from the properties of the mixed volume \(\mathbf {V}\). By this method the mixed area measure was introduced by Aleksandrov [14] and, in a similar way, by Fenchel and Jessen [348]. Whereas Aleksandrov first defined \(S(\Phi , *)\) and then used his definition of the mixed area measure to prove the expansion formula after (12.2), Fenchel and Jessen used the formula (12.1) to define \(S(\Phi , *)\) and later obtained an intuitive interpretation.

As a special case of the mixed surface area one obtains the curvature measure \(S_i(\Phi ,*)= \mathbf {S}(\Phi ,\dots ,\Phi , B,\dots , B)\), where in this formula we have *i* \(\Phi \)’s and \((n-1-i)\) *B*’s (unit balls). Thus \(S(\Phi ,*)=S_{n-1}(\Phi ,*)\). The notion of “curvature measure” was introduced by Aleksandrov [13] and Fenchel-Jessen [348], and it is a generalization of the integral of the elementary symmetric function of the principal radii of curvature to arbitrary convex bodies, as it can be defined for convex bodies with \(C^2\) boundaries. Curvature measures may also be viewed as local generalizations of the quermassintegrals, which are important in the Brunn–Minkowski Theory of convex bodies and in Integral Geometry. Nowadays there are generalizations of the curvature measures in several directions, see Schneider [1036].

The proof of Theorem

12.1.2 1) is not obvious. It was originally proved for

\(n=3\) by Minkowski in [836], using a geometric interpretation for the mixed volume of a convex polytope. To be more precise, an important ingredient of the proof is the following formula: Let

\(\Phi \) be a convex body,

*P* a polytope, and suppose

\(\sigma _1,\dots , \sigma _N\) denotes an arbitrary enumeration of the

\((n-1)\)-faces of

*P*;

\(u_i\) denotes the unit normal vectors of the face

\(\sigma _i\), and

\(v(\sigma _i)\) the

\((n-1)\)-dimensional volume of the face

\(\sigma _i\),

\(i=1,\dots , N\). Then we have:

$$\mathbf {V}(\Phi ,P,\dots , P)=\frac{1}{n}\sum _1^N\mathcal{P}_\phi (u_i)v(\sigma _i).$$

The above formula was proven for

\(n=3\) based on a procedure that was developed by Steiner [1089] for the special case of parallel bodies. We can recognize the origins of formula (

12.2) in this formula, which, as we already know, was crucial to the development of the theory. In his paper [836] Minkowski also discussed the conditions under which this equality holds true.

It is possible to characterize certain classes of convex bodies in terms of relations satisfied by mixed volumes. For example, Weil [1185] showed that an *n*-dimensional convex body \(\phi \) is contained in some translate of \(\psi \) if and only if \(\mathbf {V}(\phi , \Phi ,\dots , \Phi )\le \mathbf {V}(\psi , \Phi ,\dots , \Phi )\) for every *n*-dimensional convex body \(\Phi \). In [745], Lutwak generalized the above result by showing that the conclusion holds true if the hypothesis is satisfied only for every *n*-dimensional simplex \(\Phi \). Indeed, he derived his results from the following containment result: \(\phi \) is contained in some translate of \(\psi \) if every simplex that contains \(\phi \) also contains a translate of \(\psi \).

Godbersen’s conjecture (12.7) was independently conjectured also by Makai in [752], where he showed interesting consequences of this conjecture. Furthermore, recently Godbersen’s conjecture was almost solved by Artstein-Avidan et al. [42]. They proved that the mixed volume of *k* copies of *K* and \(n-k\) copies of \(-K\), divided by *V*(*K*), is at most \(n^n/[k^k(n-k)^{n-k}]\) which is approximately \(\left( {\begin{array}{c}n\\ k\end{array}}\right) [ 2 \pi k (n-k)/n]^{1/2} \le \left( {\begin{array}{c}n\\ k\end{array}}\right) [\pi n/2]^{1/2}\). (It seems that, using Stirling’s formula, they suppose **min**\( \{ k, n-k \} \rightarrow \infty \).) Conversely, if, e.g., *k* is less than some constant (we can say, is constant), then Godbersen’s conjecture holds up to some constant factor. This fact is already contained in Bonnesen–Fenchel [160], who proved the upper bound \(n^{\min \{ k, n-k \} }\) (and conjectured, now we know that falsely, that this would be an equality for simplices). The result of Artstein-Avidan et al. [42] gives \((e^k/k^k)n^k[1+o(1)]\) for \(n \rightarrow \infty \). For \(k \le 2\) this is asymptotically weaker than the result from [160], since then \(e^k/k^k=(e/k)^k>1^k=1\). However, for \(k \ge 3\) this is already asymptotically better than the result from [160], since then \(e^k/k^k = (e/k)^k < 1^k = 1\).

The following identity for mixed volumes holds true if \(\phi ,\psi ,\Phi _3,\dots ,\Phi _n\) are convex bodies such that \(\phi \cup \psi \) is convex; namely, then \(\mathbf {V}(\phi ,\psi ,\Phi _3,\dots ,\Phi _n) =\) \(\mathbf {V}(\phi \cup \psi ,\phi \cap \psi ,\Phi _3,\dots ,\Phi _n)\). This follows from the additivity property of mixed volumes. It was first noted by Groemer [455] and later proved in a more concise way by McMullen and Schneider [810].

The mixed volume is the unique nonnegative, multilinear, symmetric functional of convex bodies \(\mathbf {V}:(\mathcal{K}^n)^n \rightarrow \mathbb {E}\) for which \(\mathbf {V}(\Phi , \dots ,\Phi )\) is the *n*-dimensional volume \(V(\Phi )\) of \(\Phi \). Motivated by the attempt to extend the mixed volume to log-concave measures, it is interesting to characterize the classical mixed volume by some of its functional properties, not taking recourse to the notion of volume. The following properties of a function \(F:(\mathcal{K}^n)^n \rightarrow \mathbb {E}\) are certainly good candidates: *F* must be additive, increasing in each variable, and invariant under permutation of its arguments. These important properties are far from being characteristic for the mixed volume. A surprisingly mild additional vanishing condition leads to a characterization of Milman and Schneider [831], at least for centrally symmetric convex bodies. This additional condition is that \(F(\Phi _1,\dots ,\Phi _n)=0\) whenever two of the arguments \(\Phi _1,\dots ,\Phi _n\) are parallel segments. For general convex bodies only the 2-dimensional case was settled.

Another interesting question is the possibility to extend the notion of mixed volume to other kinds of argument, while preserving or generalizing its linearity properties. In [1185], Weil essentially proved that under any extension to non-convex compact sets the mixed volume would lose its essential properties.

We come now to references on mixed volumes and quermassintegrals directly referring to sets of constant width. The equations involving the quermassintegrals of bodies of constant width given in Theorem

12.1.7 are not independent of each other. Indeed, they are equivalent to

\(\frac{n+1}{2}\) equations obtained from odd values of

*k*. That is,

$$2W_{n-k}(\Phi )= \sum ^{k}_{i=0}(-1)^i \left( {\begin{array}{c}k\\ i\end{array}}\right) W_{n-i}(\Phi ) h^{k-i} \quad \text{ for } 0\le k \le n \text{ with } k \text{ odd }.$$

These relations were independently studied by Dinghas [296], Santaló [1016], and Debrunner [281]. These formulas include Blaschke’s relations of Theorem

12.1.4 for

\(n=3.\) Blaschke obtained his result by substituting

\(-h\) for

*h* in the Steiner formula for the volume of the parallel body (see Theorem

12.1.6), but without giving a complete justification for this procedure. Generalizing results of Minkowski and Blaschke, Dinghas [296] verified a recursion formula on quermassintegrals showing dependencies between different quermassintegrals of bodies of constant width and allowing the determination of all mean projection measures by only a subset of them. In view of expressing the volume of outer parallel bodies of a given convex body in terms of its integrals of principal radii of curvature, Santaló [1016] generalized Dinghas’ result and presented extensions also to inner parallel bodies; he discussed also the part of independent relations between mixed volumes for bodies of constant width (see also [235]). Namely, the quermassintegrals of sets of constant width satisfy [

\((n+1)/2\)] independent relations. For the planar case, Barbier’s Theorem

12.1.3 is obtained as a natural consequence. Debrunner [281] and Groemer [454] gave short and more elementary proofs of Dinghas’ result, and Chakerian [235] generalized Dinghas’ recursion formula for three convex bodies of constant width, one of them being the Minkowski sum of the other two convex bodies. Also in [235] he discussed how the additivity and monotonicity of mixed volumes can be used to obtain further relationships between geometric invariants of sets of constant width. Based on this he derived estimates on area and volume of constant width sets and also of rotors. A related proof from the unpublished manuscript [454] is reproduced in [238, p. 67]. Further extensions of Dinghas’ recursion formula were given by Schneider [1035], involving curvature centers (like centers of gravity and the Steiner point). More generally, in [1035] classical real-valued functions associated to convex bodies (among them also mixed volumes) are extended to vector-valued functions, where many of the well-known relations either carry over exactly, or otherwise occur in slight variants of their original forms; also bodies of constant width play a role in these investigations.

The principal ideas of the proof of the Aleksandrov–Fenchel inequality (12.5) can be traced back to the proof given by Hilbert [539] for Minkowski’s quadratic inequality (12.4) in 3-space. Aleksandrov [13] used convex bodies of class \(\mathcal{C}^2\) and Aleksandrov’s inequalities for studying mixed discriminants. There are surprising connections between the Aleksandrov–Fenchel inequality (12.5) and algebraic geometry. As we already mentioned, Khovanskií [622] and Teissier [1121] independently found algebraic approaches to the Aleksandrov–Fenchel inequality yielding a new proof via the Hodge Index Theorem, where mixed volumes of Newton polyhedra associated with Laurent polynomials play an important role. A detailed version of this proof can be found in the book by Burago and Zalgaller [196]. On the other hand, Gromov [467] continued himself this line of research by what he called a modern re-edition of Aleksandrov’s proof in which exterior products of differential forms take the place of mixed discriminants of quadratic forms. He simultaneously obtained the Hodge–Teissier–Khovanskií inequality and the Aleksandrov–Fenchel inequality.

Let *C* be a closed convex curve in the plane and *A*, *L*, *D* be its area, length, and diameter, respectively. Kubota [660] showed that for the mixed area *M* of *C* and the curve obtained from *C* by rotation through \(\pi \), the relation \(DL \ge 2(A+M)\) holds; equality is obtained only for constant width curves. In [258], the authors associated with each plane convex body a real-valued function based on the notion of mixed area and called respectively rotated mixed area. They proved convergence properties of the Fourier expansion of this function, yielding also a characterization of constant width sets.

Using the theory of mixed areas, Chakerian [230] reproved the classical Blaschke–Lebesgue theorem; with analogous methods he also derived the Minkowskian analogue (see Theorem 12.1.5). In [231], mixed volumes are represented in the framework of relative geometry, and they are used to prove properties of sets of constant relative width and constant relative brightness. Also using mixed volumes, Firey [358] reproved Minkowski’s well-known projection theorem saying that constant width is equivalent to constant girth, and in an elegant way Falconer [332] reproved it, too, see also Section 13.1. Related to this, Makai [751] showed that a \(C^2\) 3-dimensional convex body is a rotational body of constant width if \(\frac{\pi }{2}\) times its diameter equals the geodesic diameter of its boundary. Hilfssatz 1 of [1168] implies that in the above result of Makai the smoothness condition \(C^2\) can be dropped. An *n*-dimensional stability version of Minkowski’s theorem was given by Goodey and Groemer [441]. Godbersen [422] considered extensions of the Rogers–Shephard inequality (referring to the volume of a convex body and that of its difference body) to mixed volumes. He obtained related results for constant width bodies. In [1], the notion of difference body is carried over to complex vector spaces, defined via support functions depending on a gauge body. This paper contains an outstanding systematic study of geometric properties of and inequalities satisfied by the complex difference body. Among many other results, inequalities comparing quermassintegrals of difference bodies and their original bodies are proved, and the notion of constant width is also taken into consideration. Ghandehari [413] obtained several inequalities for the quermassintegrals of a convex body *K* and its polar reciprocal \(K^*\), also investigating dual mixed volumes (as defined by Lutwak in [743]), and he got inequalities for quermassintegrals of *K* and \(K^*\) if *K* is of constant width. The authors of [625] generalized Barbier’s theorem for Minkowski planes and proved a result related to mixed areas. We mention also [926], where an inequality is proved involving mixed volumes of two convex bodies \(K_1\) and \(K_2\), with \(K_2\) being centrally symmetric; equality holds iff \(K_1\) is of constant width and \(K_2\) is a ball. In [441], an inequality characterizing rotors is generalized via mixed volumes. Weakly related to constant width sets, Weissbach [1194] gave lower bounds for the quermassintegrals of centrally symmetric convex universal covers in *n*-dimensional Euclidean space, sharp for \(n=2\) and true also for the translative case of universal covers, see also Section 15.2.