On Hodge-Riemann relations for translation-invariant valuations
aa r X i v : . [ m a t h . M G ] S e p ON HODGE-RIEMANN RELATIONSFOR TRANSLATION-INVARIANT VALUATIONS
JAN KOTRBAT ´Y
Abstract.
The Alesker product turns the space of smooth translation-invariant valuations onconvex bodies into a commutative associative unital algebra, satisfying Poincar´e duality andthe hard Lefschetz theorem. In this article, a version of the Hodge-Riemann relations for theAlesker algebra is conjectured, and the conjecture is proved in two particular situations: foreven valuations, and for 1-homogeneous valuations. The latter result is then used to deducea special case of the Aleksandrov-Fenchel inequality. Finally, mixed versions of the hard Lef-schetz theorem and of the Hodge-Riemann relations are conjectured, and it is shown that theAleksandrov-Fenchel inequality follows from the latter in its full generality. Introduction A valuation is—at least for the purpose of the text that follows—a finitely additive measureon convex bodies in the Euclidean space. Of particular interest is the space Val of valuationsthat are continuous and translation invariant; it was conjectured by Peter McMullen [40] thatVal is just the weak completion of the linear space of mixed volumes . McMullen’s conjecture wasproved by Semyon Alesker [4], in fact, in much greater generality of what has become known asthe irreducibility theorem . The salient feature of Alesker’s result is that it acted as a catalyst fora variety of further developments, among the most important of which is certainly the discoveryof a natural product of valuations.The Alesker product [7] is defined on the dense subspace Val ∞ ⊂ Val of smooth valuations(we refer to § n of the underlying Euclidean space.Remarkably, the Alesker algebra moreover satisfies a Poincar´e-type duality , and even a version ofthe hard Lefschetz theorem for multiplication by the first intrinsic volume. Dual to the productis another natural multiplicative structure on Val ∞ , the Bernig-Fu convolution introduced in[20]. The product and the convolution are further intertwined by a
Fourier-type transform ,which was constructed by S. Alesker in [5, 8]. It is apropos to point out that all these structuresbehave naturally with respect to the ‘standard’ examples of valuations, such as to the Eulercharacteristic and the Lebesque measure, or, more generally, to the intrinsic and mixed volumes.A fundamental connection between the product of valuations and integral geometry wasfound by Bernig and Fu [20, 31]: It turns out that the array of classical kinematic formulas forthe intrinsic volumes—to which great attention was paid throughout the twentieth century, inparticular, in works of Blaschke, Chern, Hadwiger, Federer, or Santal´o—precisely correspondsto the structure of the (finite-dimensional) subalgebra Val
SO( n ) ⊂ Val ∞ of SO( n )-invariantvaluations. Furthermore, the same correspondence applies to any closed subgroup G ⊂ SO( n )acting transitively on spheres, and can even be extended to a general isotropic space [10].Application of this principle resulted in a comprehensive understanding of Hermitian integralgeometry [16, 21, 22, 52, 53], and led to substantial progress in the other—quaternionic andoctonionic—spaces [17, 24, 39, 47]. Date : September 2, 2020.1991
Mathematics Subject Classification. .1. Main results.
The aim of this article is to explore a new, yet very natural feature of thealgebra of smooth valuations. Our results indicate that Alesker’s algebraic theory of valuationsis even more powerful and encompassing than so far believed. Specifically, we show that not onlyit explains and extends integral geometry of intrinsic volumes, but it also appears capable ofsubsuming another main pillar of classical convex geometry—the Aleksandrov inequalities—andthus merging these two seemingly separated topics into a unified concept.To motivate our work, let us recall a fundamental result from the cohomology theory ofcompact K¨ahler manifolds that is closely related to the Poincar´e duality (PD) and the hardLefschetz theorem (HL): A legitimate question to ask is what is the signature of the Hermitianpairing obtained by composing the Poincar´e pairing with the Lefschetz map. Since the latter isself-adjoint with respect to the former, it is enough to restrict the problem to primitive elements.According to the
Hodge-Riemann relations (HR), the pairing is then either positive or negativedefinite, depending on the degree of the cohomology class (see, e.g., § K¨ahler package
PD–HL–HR has appeared in numerous differentsituations outside of K¨ahler geometry [1, 25, 29, 34, 36, 42], and very often proved to be the keyto solving a long-standing open problem (see also Huh’s lucid account [33]).In this article, a step is taken towards completing the K¨ahler package for the algebra Val ∞ ;namely, a version of the Hodge-Riemann relations is proved in two particular situations.In order to recast the above problem in the language of the Alesker product, let us fixsome notation: First, let Val ∞ k ⊂ Val ∞ denote the subspace of k -homogeneous elements, andlet µ ∈ Val ∞ be the first intrinsic volume. Further, assume 0 ≤ k ≤ ⌊ n ⌋ , where n is thedimension of the underlying Euclidean space, and let us call a valuation φ ∈ Val ∞ k primitive if φ · µ n − k +11 = 0. Finally, consider the Alesker-Hodge-Riemann pairing Q : Val ∞ k × Val ∞ k → Val ∞ n given by Q ( φ, ψ ) = φ · ψ · µ n − k . (1)After the standard identification Val ∞ n ∼ = C , Q becomes an (a priori non-degenerate) Hermitianform. As far as its signature on primitive elements is concerned, a careful look at the variousknown subalgebras of G -invariant valuations—in particular, at the sufficiently complicated case G = Spin(9), resolved recently in author’s Ph.D. thesis [39]—shows that it seems to preciselyreflect the parity of the degree k . Since all such valuations are in fact even (curiously, evenwhen − id / ∈ G ), it is natural to expect a more general phenomenon underlying the behaviour,at least in this case. Indeed, it is our first main result that Theorem A.
Let ≤ k ≤ ⌊ n ⌋ . Any non-zero primitive even valuation φ ∈ Val ∞ k satisfies ( − k Q ( φ, φ ) > . (2)The general situation turns out to be slightly more subtle: Perhaps surprisingly, already in thecase k = 1, which is clearly the first non-trivial one to deal with as 0-homogeneous valuationsare constant and thus even, the dependence of the signature of the Alesker-Hodge-Riemannpairing on the parity of a valuation starts manifesting itself. Specifically, let even valuationshave parity 0, and odd valuations parity 1. The second main result of our article is then asfollows: Theorem B.
Let s ∈ { , } . Any non-zero primitive valuation φ ∈ Val ∞ of parity s satisfies ( − s Q ( φ, φ ) > . (3)The proofs of Theorems A and B both rely on representing the valuations in question bysmooth functions (on Grassmannians in the former case, on the sphere in the latter). In eithersituation the Lefschetz map boils down to a certain SO( n )-equivariant integral transform whichmakes it possible to employ representation theory and to deduce the result from the sign of itseigenvalues. In the case of 1-homogeneous valuations, a large part of the work has already beendone by Bernig and Hug in their recent article [23], upon which our proof is heavily based. .2. Conjectures.
Although the techniques behind the two main results of this article certainlyreach their limits here and, therefore, different arguments would be needed in order to prove astronger statement, we believe it is reasonable to expect the Hodge-Riemann relations to hold ina more general context. Let us formulate precise conjectures. First, the natural generalizationof Theorems A and B would certainly be as follows:
Conjecture C.
Let ≤ k ≤ ⌊ n ⌋ and s ∈ { , } . Any non-zero primitive valuation φ ∈ Val ∞ k of parity s satisfies ( − k + s Q ( φ, φ ) > . (4)Second, applying the Alesker-Fourier transform, the Hodge-Riemann-type relations (4) canbe reformulated in terms of the Bernig-Fu convolution. In this case, the algebra is graded bythe co-degree, and the Lefschetz map is induced by the pre-last intrinsic volume µ n − ∈ Val ∞ n − .Let us say that a valuation φ ∈ Val ∞ n − k is co-primitive if φ ∗ µ ∗ ( n − k +1) n − = 0, and consider the Bernig-Fu-Hodge-Riemann pairing e Q : Val ∞ n − k × Val ∞ n − k → Val ∞ ∼ = C given by e Q ( φ, ψ ) = φ ∗ ψ ∗ µ ∗ ( n − k ) n − . (5)We show that Conjecture C is equivalent to Conjecture D.
Let ≤ k ≤ ⌊ n ⌋ . Any non-zero co-primitive valuation φ ∈ Val ∞ n − k satisfies ( − k e Q ( φ, φ ) > . (6)Notice that this setting appears to be the more natural one as (6) no longer depends on theparity; we shall also see that it is more appropriate for applications and further extensions.Further, let us point out that Conjecture D is true in the proven cases of Conjecture C, i.e., foreven valuations or if k = 1.Very often the K¨ahler package is as powerful as proclaimed only if it comes in a more generalversion than we have discussed here. Namely, instead of being the composition of several copiesof the same map, the Lefschetz map (in both HL and HR) is considered to be composed of generalelements of a certain cone of operators. For K¨ahler manifolds, in fact, such mixed versions ofHL and HR were proved only recently by Dinh and Nguyˆen [28] who completed earlier work ofGromov [32]. Motivated by these developments, we, finally, conjecture the mixed versions of thehard Lefschetz theorem and of the Hodge-Riemann relations for the algebra Val ∞ as follows: Conjecture E.
Let ≤ k ≤ ⌊ n ⌋ and let K k , . . . , K n be convex bodies with smooth boundaryand positive curvature. (a) The mapping
Val ∞ n − k → Val ∞ k given by φ φ ∗ V ( · [ n − , K k +1 ) ∗ · · · ∗ V ( · [ n − , K n )(7) is an isomorphism. (b) Let φ ∈ Val ∞ n − k be non-zero and such that φ ∗ V ( · [ n − , K k ) ∗ V ( · [ n − , K k +1 ) ∗ · · · ∗ V ( · [ n − , K n ) = 0 . (8) Then ( − k φ ∗ φ ∗ V ( · [ n − , K k +1 ) ∗ · · · ∗ V ( · [ n − , K n ) > . (9)Observe that the two parts of Conjecture E generalize their non-mixed counterparts Theorem2.7 below and Conjecture D, respectively, since µ n − is positively proportional to the mixedvolume V ( · [ n − , D ), with D being the Euclidean ball. Let us also point out that reformulatingthe previous conjecture back in terms of the Alesker product would involve—at least under theadditional assumption of central symmetry—the so-called Holmes-Thompson intrinsic volumes ,i.e., valuations naturally assigned to a general smooth Minkowski space (cf. [13, 15, 30]). .3. Applications to inequalities of geometric type.
It turns out that some of the purelyalgebraic formulations of the aforedescribed results and conjectures can in fact acquire a geo-metric meaning. This finally provides the anticipated connection between the algebraic theoryof valuations and the isoperimetric inequalities.First, we show that Conjecture D—used in the situation k = 1 where we know it is true—yields an important special case of the Aleksandrov-Fenchel inequality, namely, Minkowski’ssecond inequality for one body being the Euclidean ball, or, in other words, the isoperimetricinequality between the first and the second intrinsic volume. This follows a simple argument, aspecial case of which was communicated to us by S. Alesker.Furthermore, using a ‘mixed’ version of the same argument, we show that the correspondingspecial case of the conjectured mixed Hodge-Riemann relations (Conjecture E)—if true—yieldthe Aleksandrov-Fenchel inequality in its full generality. In particular, this would establish alsothe rest of the isoperimetric inequalities. Recall that one way to formulate these is to say thatthe sequence of (properly normalized) intrinsic volumes is log-concave. To quote June Huh [33], the log-concavity of a sequence is not only important because of its applications but because ithints the existence of a structure that satisfies PD, HL, and HR . Conjecture E may thus beviewed as yet another argument in favour of this phenomenon, and vice versa.To conclude the introduction, let us point out that among the numerous known proofs ofthe Aleksandrov-Fenchel inequality [2, 3, 27, 42, 46, 51] (see also [26], § −∞ of generalized translation-invariant valuations in which Val ∞ is densely embedded. Itwould certainly be interesting to know whether Conjecture E admits some generalization toVal −∞ from which McMullen’s results [42] could be eventually recovered. Acknowledgements.
I am grateful to Karim Adiprasito and Fr´ed´eric Chapoton for extremelyvaluable discussions out of which this project actually arose; to Semyon Alesker for his constantinterest in this work and for numerous helpful suggestions, in particular, for the beautiful ideaof applying the results to geometric-type inequalities; to the DAAD Foundation that supportedmy stay at the Tel Aviv University where this work was initiated; as well as to Franz Schusterand Thomas Wannerer for their encouragement and many useful comments on earlier versionsof the paper. 2.
Preliminaries
Valuations on convex bodies.
Let us begin by recalling some necessary facts from bothclassical and modern theory of valuations. Our references for this and the following section aremonographs [9] and [44].To establish notation, let K be the family of convex bodies, i.e., non-empty compact convexsets in R n . On this space, the operations of Minkowski addition and scalar multiplication are,respectively, given for K, L ∈ K and α ∈ R as follows: K + L = { x + y ; x ∈ K, y ∈ L } , (10) αK = { αx ; x ∈ K } . (11)A functional φ : K → C is called a valuation if φ ( K ∪ L ) + φ ( K ∩ L ) = φ ( K ) + φ ( L )(12)holds for any K, L ∈ K with K ∪ L ∈ K . A valuation φ is said to be translation invariant if φ ( K + { x } ) = φ ( K ) holds for any K ∈ K and x ∈ R n , and continuous if it is so with respectto the Hausdorff metric ρ H ( K, L ) = inf { ε > K ⊂ L + εD, L ⊂ K + εD } , where D ⊂ R n isthe (origin-centered) unit Euclidean ball. We shall consider entirely valuations enjoying boththese properties and denote the vector space of all such by Val. Let further Val k ⊂ Val be the ubspace of valuations satisfying φ ( λK ) = λ k φ ( K ) for any λ > K ∈ K . Remarkably, onehas the McMullen grading Val = n M k =0 Val k , (13)in consequence of which Val becomes a Banach space with respect to k φ k = sup K ⊂ D | φ ( K ) | .Another, obvious, grading is with respect to parity:Val = Val ⊕ Val , (14)where Val s = { φ ∈ Val ; φ ( − K ) = ( − s φ ( K ) for any K ∈ K} . We denote Val sk = Val k ∩ Val s .There is a natural GL( n, R ) action on Val given by ( g, φ ) φ ◦ g − . Clearly, each Val sk is thena closed invariant subspace. The following result is central to the theory of valuations: Theorem 2.1 (Alesker [4]) . For any ≤ k ≤ n and s = 0 , , the GL( n, R ) module Val sk isirreducible, i.e., admits no proper closed invariant subspace. Observe that the statement is only non-trivial for 0 < k < n . Indeed, one has Val = span { χ } ,where χ ≡ n = span { vol n } is spanned by theLebesgue measure. The former is easy to see, while the latter is a deep result of Hadwiger. Asit is usual, let us identify Val and Val n with C via χ and vol n , respectively.A broad generalization of the two prime examples is provided by the concept of mixed volumes .Recall that the mixed volume of K , . . . , K n ∈ K is defined as V ( K , . . . , K n ) = 1 n ! ∂ n ∂λ · · · ∂λ n (cid:12)(cid:12)(cid:12)(cid:12) λ ,...,λ n =0 vol n ( λ K + · · · + λ n K n ) . (15)According to a classical result of Minkowski, the function being differentiated on the right-handside of (15) is actually a homogeneous polynomial of degree n . It is well known that V is totallysymmetric, real valued, and non-negative, and that V ( · [ k ] , K k +1 , . . . , K n ) ∈ Val k for any K i ,where [ k ] stands for k copies. In particular, µ k = c k V ( · [ k ] , D [ n − k ])(16)is the k -th intrinsic volume . The exact values of the normalizing constants c k > µ = χ and µ n = V ( · [ n ]) = vol n .Famously, the intrinsic volumes satisfy, for any K ∈ K , the Aleksandrov- or isoperimetricinequalities µ ( K ) µ ( D ) ≥ (cid:18) µ ( K ) µ ( D ) (cid:19) ≥ · · · ≥ (cid:18) µ k ( K ) µ k ( D ) (cid:19) k ≥ · · · ≥ (cid:18) µ n ( K ) µ n ( D ) (cid:19) n . (17)In fact, (17) is only a consequence of a much stronger result, namely, the Aleksandrov-Fenchelinequality for mixed volumes: For all K , . . . , K n ∈ K , one has V ( K , K , K , . . . , K n ) ≥ V ( K , K , K , . . . , K n ) V ( K , K , K , . . . , K n ) . (18)According to McMullen’s conjecture [40], whose proof is an easy consequence of Theorem 2.1,mixed volumes span a dense subset of Val. By the standard polarization formulas, the same istrue for valuations of the form vol n ( · + K ), with K ∈ K .2.2. Smooth valuations.
A valuation φ ∈ Val is said to be smooth if the Banach-space-valuedmapping GL( n, R ) → Val given by g φ ◦ g − is infinitely differentiable. It is a general factfrom representation theory that the subspace Val ∞ ⊂ Val of such valuations is invariant, dense,and carries a natural Fr´echet-space topology (stronger than that induced from Val), with whichVal ∞ will be tacitly assumed to be endowed. In this connection, observe that the irreducibilitytheorem (Theorem 2.1) holds verbatim for the GL( n, R ) modules Val s, ∞ k = Val sk ∩ Val ∞ as wellas the gradings (13) and (14) for, respectively, Val ∞ k = Val k ∩ Val ∞ and Val s, ∞ = Val s ∩ Val ∞ . he notion of smoothness is essential to the modern valuation theory as Val ∞ can be naturallyequipped with an array of striking algebraic structures that do not, however, extend to Val.To this end, let us recall that the valuations V ( · [ k ] , K k +1 , . . . , K n ) and vol n ( · + K ) are smoothprovided K i and K , respectively, belong to the class K ∞ + of convex bodies with non-emptyinterior, smooth boundary, and positive Gauss-Kronecker curvature. In particular, the intrinsicvolumes are such. Against this background, the Alesker product and the
Bernig-Fu convolution are, respectively, given as follows:
Theorem 2.2 (Alesker [7]) . Let ∆ : R n → R n : x ( x, x ) be the diagonal embedding. Thereexists a unique bilinear, continuous product · on Val ∞ such that vol n ( · + K ) · vol n ( · + L ) = vol n (∆( · ) + K × L )(19) holds for any K, L ∈ K ∞ + . Moreover, (a) · is commutative and associative, (b) φ · χ = φ for any φ ∈ Val ∞ , (c) Val s, ∞ k · Val r, ∞ l ⊂ Val q, ∞ k + l with q = r + s mod 2 , (d) µ k · µ l = c k,l µ k + l for some c k,l > , (e) the product pairing Val ∞ k × Val ∞ n − k → Val ∞ n ∼ = C is non-degenerate, i.e., for any non-zero φ ∈ Val ∞ k there is ψ ∈ Val ∞ n − k with φ · ψ = 0 . Theorem 2.3 (Bernig, Fu [20]) . There exists a unique bilinear, continuous product ∗ on Val ∞ such that vol n ( · + K ) ∗ vol n ( · + L ) = vol n ( · + K + L )(20) holds for any K, L ∈ K ∞ + . Intertwining the product and the convolution is the so-called
Alesker-Fourier transform : Theorem 2.4 (Alesker [8]) . There exists a canonical isomorphism F : Val ∞ → Val ∞ of Fr´echetspaces such that (a) F ( φ · ψ ) = F φ ∗ F ψ holds for any φ, ψ ∈ Val ∞ , (b) F Val s, ∞ k = Val s, ∞ n − k , (c) F = ( − s id on Val s, ∞ , (d) F µ k = µ n − k . Synthesis of Theorems 2.2 and 2.4 yields at ones
Corollary 2.5. (a) ∗ is commutative and associative, (b) φ ∗ vol n = φ for any φ ∈ Val ∞ , (c) Val s, ∞ n − k ∗ Val r, ∞ n − l ⊂ Val q, ∞ n − k − l with q = r + s mod 2 , (d) µ n − k ∗ µ n − l = c k,l µ n − k − l for some c k,l > , (e) for any non-zero φ ∈ Val ∞ k there is ψ ∈ Val ∞ n − k with φ ∗ ψ = 0 . Parts (e) of Theorem 2.2 and Corollary 2.5 are versions of
Poincar´e duality for valuations.The following properties of the convolution will be also useful for us:
Lemma 2.6 (Bernig, Fu [20]) . (a) Let k + l ≤ n . There is a constant c k,l > such that for any K , . . . , K k , L , . . . , L l ∈ K ∞ + , V ( · [ n − k ] , K , . . . , K k ) ∗ V ( · [ n − l ] , L , . . . , L l )= c k,l V ( · [ n − k − l ] , K , . . . , K k , L , . . . , L l ) . (21)(b) For any φ ∈ Val ∞ and K ∈ K , one has ( µ n − ∗ φ )( K ) = 12 ddλ (cid:12)(cid:12)(cid:12)(cid:12) λ =0 φ ( K + λD ) . (22) inally, as discussed in the introduction, two versions of the hard Lefschetz theorem hold forsmooth valuations: Theorem 2.7 (Bernig, Br¨ocker [18]) . Let ≤ k ≤ ⌊ n ⌋ . The mapping Val ∞ n − k → Val ∞ k given by φ φ ∗ µ ∗ ( n − k ) n − (23) is an isomorphism of Fr´echet spaces. Corollary 2.8.
Let ≤ k ≤ ⌊ n ⌋ . The mapping Val ∞ k → Val ∞ n − k given by φ φ · µ n − k (24) is an isomorphism of Fr´echet spaces. Observe that it follows at once from (19) that( φ ◦ g − ) · ( ψ ◦ g − ) = ( φ · ψ ) ◦ g − (25)holds for any φ, ψ ∈ Val ∞ and g ∈ GL( n, R ), and the same applies to the Bernig-Fu convolution.In particular, both the product with µ and the convolution with µ n − commute with themaximal compact subgroup O( n ) of GL( n, R ), as the intrinsic volumes are invariant under it.2.3. Functions on Grassmannians and integral transforms.
Let us now collect someknown facts about the Hilbert space of square-integrable functions on the Grassmann manifoldGr k of k -dimensional real subspaces in R n . Enough for us will be to assume 0 ≤ k ≤ ⌊ n ⌋ .First of all, L (Gr k ) has an obvious SO( n )-module structure. In this connection, recall thatthe family of irreducible SO( n ) representations is parametrized by the set Λ of their highestweights, where, if n = 2 m + 1 is odd,Λ = { ( λ , . . . , λ m ) ∈ Z m ; λ ≥ λ ≥ · · · ≥ λ m ≥ } , (26)while for n = 2 m even,Λ = { ( λ , . . . , λ m ) ∈ Z m ; λ ≥ λ ≥ · · · ≥ λ m − ≥ | λ m |} . (27)In both cases, for 0 ≤ k ≤ m = ⌊ n ⌋ , let us denoteΛ k = Λ ∩ (cid:16) (2 Z ) k × { } m − k (cid:17) . (28)It is well known from works of Sugiura [49] and Takeuchi [50] (see also [48]) that the decompo-sition of L (Gr k ) into irreducible SO( n ) modules is multiplicity free. More precisely, L (Gr k ) = [ M λ ∈ Λ k H λ , (29)where H λ ⊂ C ∞ (Gr k ) is the irreducible SO( n )-module corresponding to the highest weight λ ,and cL denotes the ( L -orthogonal) Hilbert-space direct sum (cf. [45], § λ = (2 m , . . . , m k , , . . . , ∈ Λ k , the highest-weight vector h λ ∈ H λ (which isunique up to scaling) was described by Strichartz [48]: Assume first k ≥
1. Take an arbitrary E ∈ Gr k and let x , ... x n, , . . . , x ,k ... x n,k be its orthonormal basis. For 1 ≤ l ≤ k , we first define A [ l ] = x , + √− x , · · · x ,k + √− x ,k x , + √− x , · · · x ,k + √− x ,k ... ... x l − , + √− x l, · · · x l − ,k + √− x l,k ∈ R l × k . (30) hen, if m k ≥
0, one puts h λ ( E ) = k − Y l =1 det (cid:16) A [ l ] A [ l ] t (cid:17) m l − m l +1 det (cid:16) A [ k ] A [ k ] t (cid:17) m k , (31)and, if m k < h λ ( E ) = k − Y l =1 det (cid:16) A [ l ] A [ l ] t (cid:17) m l −| m l +1 | det ( A [ k ] A [ k ] t ) | m k | . (32)It is straightforward to verify that this definition is independent of the choice of the orthonormalbasis. If k = 0, one necessarily has λ = (0 , . . . ,
0) and defines h λ ( { } ) = 1.Let us recall two important integral transforms on Grassmanians. First, for 0 ≤ k ≤ l ≤ n ,we define the Radon transform R k,l : L (Gr k ) → L (Gr l ) as follows:( R k,l f )( E ) = Z Gr k ( E ) f ( F ) dF, (33)where Gr k ( E ) = { F ∈ Gr k ; F ⊂ E } . Notice that R k,k = id. Reversing the inclusion F ⊂ E ,one can extend the definition of the Radon transform also to k > l . For our purpose, however,the above form will be sufficient.Second, consider E, F ∈ Gr k , 0 ≤ k ≤ n . Take any compact subset A ⊂ E with vol k ( A ) = 0and put | cos( E, F ) | = vol k ( π F A )vol k ( A ) , (34)where π F : R n → F is the orthogonal projection. Notice that (34) is independent of the choiceof the set A . Then the cosine transform T k : L (Gr k ) → L (Gr k ) is defined as( T k f )( E ) = Z Gr k | cos( E, F ) | f ( F ) dF. (35)Observe that both R k,l and T k are linear, continuous, and SO( n ) equivariant, in consequenceof which they map smooth functions to smooth functions. In general, one can define the cosinetransform between functions on Grassmanians of distinct rank, or one can consider the so-called α -cosine transform where the kernel is raised to α ∈ C . Again, such generalizations willnot be considered here. What we shall need instead are known values of the cosine-transformmultipliers: Let ⊥ : Gr k → Gr n − k be the operation of taking the orthogonal complement, andfor ν ∈ C and k ∈ N , consider the Pochhammer symbol( ν ) k = k − Y j =0 ( ν + j ) . (36) Lemma 2.9 (Zhang [54], see also [12]) . Let ≤ k ≤ ⌊ n ⌋ and λ = (2 m , . . . , m k , , . . . , ∈ Λ k .Then, for some normalizing constant c k > , ⊥ ∗ ◦ T n − k ◦ ⊥ ∗ | H λ = c k k − Y j =1 (cid:16) j − m j (cid:17) m j (cid:16) n − j (cid:17) m j (cid:16) k − | m k | (cid:17) | m k | (cid:16) n − k (cid:17) | m k | id | H λ . (37)2.4. Functional representations of even valuations.
There are two important ways torepresent even smooth valuations in terms of smooth functions on Grassmanians. First, the
Crofton map
Cr : C ∞ (Gr k ) → Val , ∞ k is given byCr( f )( K ) = Z Gr k f ( E ) vol k ( π E K ) dE, f ∈ C ∞ (Gr k ) , K ∈ K . (38)It is readily verified that the map is indeed well defined and linear (cf. [11]). Importantly, heorem 2.10 (Alesker, Bernstein [11], see also [5]) . Let ≤ k ≤ ⌊ n ⌋ . The following restric-tion of the Crofton map is an isomorphism: Cr : C ∞ (Gr k ) ∩ [ M λ ∈ Λ k | λ |≤ H λ → Val , ∞ k . (39)Let us point out that since the isomorphism commutes with the natural action of SO( n ),(39) also gives the decomposition of Val , ∞ k into irreducible SO( n ) modules. In the sense of thepreceding theorem, we shall always denote f φ = Cr − ( φ ) , φ ∈ Val , ∞ k . (40)In other words, each φ ∈ Val , ∞ k will be represented uniquely by means of φ ( K ) = Z Gr k f φ ( E ) vol k ( π E K ) dE, K ∈ K . (41)Second, the Klain map
Kl : Val , ∞ k → C ∞ (Gr k ) is defined as follows: According to Hadwiger’scharacterization theorem, the restriction of any φ ∈ Val , ∞ k to an arbitrary k -plane E ∈ Gr k must be a multiple of the Lebesgue measure vol k on E . Then, one defines Kl( φ ) = Kl φ by φ | E = Kl φ ( E ) vol k . (42)It can be shown that this is a well-defined, linear, and SO( n )-equivariant mapping. Importantly,it was shown by Klain [37] that Kl is injective. As a side note, one has Kl ◦ Cr = T k on C ∞ (Gr k ).We shall need a few more results concerning the expression of some of the algebraic structuresintroduced in § φ ∈ Val , ∞ is determined by f F φ = ⊥ ∗ f φ (43)(see [5, 8]). Second, the Lefschetz map boils down to Lemma 2.11 (Alesker [6]) . Let ≤ k + l ≤ n . There is c k,l > such that for any φ ∈ Val , ∞ k , Kl φ · µ l = c k,l T k + l ◦ R k,k + l ( f φ ) . (44)Let us point out that the constant c k,l was only specified to be non-zero in the original versionof the previous lemma as formulated in [6]. However, it is easy to trace back through Alesker’sproof to see that c k,l is indeed real and positive . Finally, as for the Alesker-Poincar´e pairing, Lemma 2.12 (Bernig, Fu [20]) . For ϕ, ψ ∈ Val , ∞ k one has F ϕ · ψ = Z Gr k f ϕ ( E ) Kl ψ ( E ) dE. (45)2.5. Spherical valuations.
There is another broad class of smooth valuations representedfaithfully by means of functions, this times on the unit sphere. Recall that L ( S n − ) = [ M q ∈ N S q , (46)where S q ⊂ C ∞ ( S n − ) is the space of spherical harmonics, i.e. restrictions of q -homogeneousharmonic polynomials on R n . If f ∈ L ( S n − ) is smooth, then the expansion (46) convergesalso with respect to the standard Fr´echet-space structure on C ∞ ( S n − ) (see [43], § ≤ k ≤ n −
1, let further S k ( K, · ) be the k -th area measure of a convex body K ∈ K (we refer to § ≤ k ≤ n −
1, there are constants c k,j > S k ( K + λD, ω ) = k X j =0 c k,j λ j S k − j ( K, ω )(47) olds for any λ > K ∈ K , and any Borel subset ω ⊂ S n − . It is easy to see that for a smoothfunction f ∈ C ∞ ( S n − ), the valuation defined by µ k,f ( K ) = Z S n − f ( y ) dS k ( K, y ) , K ∈ K , (48)belongs to Val ∞ k . Valuations of the form (48) are called spherical . Observe that, in particular,the intrinsic volumes are such, as µ k, is proportional to µ k . In this connection, let us take theliberty of assuming that the area measures are normalized such that µ k, = µ k . (49)Notice also that µ k,f = 0 provided f ∈ S is the restriction of a linear functional, and that µ k,f is even/odd if the same is true for f .The following recent results concerning spherical valuations will be useful for us: Lemma 2.13 (Bernig, Hug [23]) . Let ≤ k ≤ n − and q, r ∈ N with q, r = 1 . (a) There is a constant c k,q > such that for any f ∈ S q , F µ k,f = c k,q ( √− q µ n − k,f . (50)(b) There is a constant ˜ c k,q > such that for any f ∈ S q and g ∈ S r , µ k,f · µ n − k,g = ˜ c k,q ( − q (cid:16) − q ( n + q − n − (cid:17) R S n − f ( y ) g ( y ) dy if q = r, otherwise. (51) Lemma 2.14 (Alesker [14]) . Let C ∞ ( S n − ) ⊂ C ∞ ( S n − ) be the Fr´echet subspace of functionswhose orthogonal projection to S is trivial. Then the mapping C ∞ ( S n − ) → Val ∞ given by f µ ,f (52) is an isomorphism of Fr´echet spaces. Primitive valuations, the Lefschetz decomposition for valuations,and the Alesker-Hodge-Riemann pairing
Let 0 ≤ k ≤ ⌊ n ⌋ . We say that a k -homogeneous smooth valuation φ ∈ Val ∞ k is primitive if φ · µ n − k +11 = 0 , (53)and denote the subspace of all such valuations by P k . We shall also use the following naturalnotation: P sk = P k ∩ Val s, ∞ , s = 0 ,
1. In these terms, an easy consequence of the hard Lefschetztheorem is a version of the
Lefschetz decomposition for valuations. Namely,
Corollary 3.1.
For any ≤ k ≤ ⌊ n ⌋ , one has Val ∞ k = k M j =0 µ k − j · P j , (54) and consequently, Val s, ∞ k = k M j =0 µ k − j · P sj , s = 0 , . (55) Proof.
Since the multiplication by µ commutes with − id, (55) is an immediate consequence of(54) in fact. Let us thus show the latter. The case k = 0 is trivial so assume otherwise.Obviously, µ · Val ∞ k − + P k ⊂ Val ∞ k . The opposite inclusion follows from the surjectivity partof the hard Lefschetz theorem (Corollary 2.8) in degree k −
1: For any φ ∈ Val ∞ k , there exists ψ ∈ Val ∞ k − with µ n − k +21 · ψ = µ n − k +11 · φ . Then φ = µ · ψ + ( φ − µ · ψ ) yields the desireddecomposition. Finally, if φ ∈ P k ∩ µ · Val ∞ k − , then 0 = µ n − k +11 · φ = µ n − k +21 · ψ for some ψ ∈ Val ∞ k − which must be trivial by the injectivity part of the hard Lefschetz theorem. All inall, we have Val ∞ k = µ · Val ∞ k − ⊕ P k from which the rest follows easily by induction. (cid:3) n order to proceed to discuss the Hodge-Riemann relations in the algebra Val ∞ , let us recallfrom the introduction that the Alesker-Hodge-Riemann pairing Q : Val ∞ k × Val ∞ k → C is givenby Q ( φ, ψ ) = φ · ψ · µ n − k , (56)where Val n is identified with C as usual.4. Hodge-Riemann relations for even valuations
In the section to follow, the Hodge-Riemann relations are proved for even smooth valuations.Our proof relies on representing the valuations in question by smooth functions, and the involvedalgebraic structures in terms of the Radon and cosine transforms, as summarized in § § § ≤ k ≤ ⌊ n ⌋ . Since the Lefschetz map, i.e., the multiplication by µ , is SO( n ) equivariant,it follows at once from the Lefschetz decomposition (55) that the SO( n ) module P k composesprecisely of irreducible subspaces corresponding to the following highest weights:Π k = n ( m , . . . , m k , , . . . , ∈ Λ k ; | m | ≤ , m k = 0 o . (57)Since the Crofton map is equivariant as well, this together with Theorem 2.10 means thatCr − (P k ) = C ∞ (Gr k ) ∩ [ M λ ∈ Π k H λ . (58)Later on, we shall need to control the sign of the Radon-transform eigenvalues correspondingto the highest weights Π k . To this end, Lemma 4.1.
Let ≤ k ≤ ⌊ n ⌋ and λ = (2 m, , . . . , , ± , , . . . , ∈ Π k . There exists c k,m > such that ⊥ ∗ ◦ R k,n − k | H λ = c k,m ( − m − k id | H λ . (59) Remark 4.2.
Observe that the last non-zero entry of λ is the k -th one. Proof.
By Schur’s Lemma, there is γ ∈ C such that ⊥ ∗ ◦ R k,n − k | H λ = γ id | H λ . (60)Let e j be the j -th element of the standard orthonormal basis of R n . Consider the highest-weightvector h λ ∈ H λ and E ∈ Gr k spanned by the (orthonormal) basis { e , e , e , . . . , e k − } . Then A [1] = (1 , , . . . , , and A [ k ] = id k . Therefore, h λ ( E ) = det (cid:16) A [1] A [1] t (cid:17) m − det (cid:16) A [ k ] A [ k ] t (cid:17) = 1 , and, consequently, one has γ = [ ⊥ ∗ ◦ R k,n − k ( h λ )] ( E ) = Z Gr k ( E ⊥ ) h λ ( F ) dF. (61) onsider an arbitrary F ∈ Gr k ( E ⊥ ). Any orthonormal basis of F must be of the following form: x , x , ...0 x k, ∗ , . . . , x ,k x ,k ...0 x k,k ∗ . Here ‘ ∗ ’ stands for the remaining part of a vector which is insignificant to us. For such a basis, A [1] = √− x , , x , , . . . , x ,k ) ∈ R × k , and A [ k ] = √− X, where X = x , · · · x ,k ... ... x k, · · · x k,k ∈ R k × k . Hence, for some c k,m,F ≥ h λ ( F ) = det (cid:16) A [1] A [1] t (cid:17) m − det (cid:16) A [ k ] A [ k ] t (cid:17) = ( − m − k X j =1 ( x ,j ) m − ( − k (det X ) = c k,m,F ( − m − k . Plugging this into (61) and taking into account that h λ is continuous and that for F = R { e , e , . . . , e k } ∈ Gr k ( E ⊥ )one has h λ ( F ) = ( − m − k = 0 , we finally obtain that, for some c k,m > γ = c k,m ( − m − k . (cid:3) As no similarly simple argument is known to us to prove a counterpart statement for the cosinetransform, we deduce it as an consequence of the explicit general result of Zhang (Lemma 2.9).
Lemma 4.3.
Let ≤ k ≤ ⌊ n ⌋ and λ = (2 m, , . . . , , ± , , . . . , ∈ Π k . There exists c k,m > such that ⊥ ∗ ◦ T n − k ◦ ⊥ ∗ | H λ = c k,m ( − m − id | H λ . (62) Proof.
We apply Lemma 2.9 for m = m and m = · · · = m k − = | m k | = 1. In this case, k − Y j =1 (cid:16) j − m j (cid:17) m j (cid:16) n − j (cid:17) m j (cid:16) k − | m k | (cid:17) | m k | (cid:16) n − k (cid:17) | m k | = (cid:16) − m (cid:17) m (cid:16) + n (cid:17) m k Y j =2 j n − j . Since 1 (cid:16) + n (cid:17) m k Y j =2 j n − j > nd, for some c m > (cid:18) − m (cid:19) m = (cid:18) − m (cid:19) (cid:18) − m + 1 (cid:19) · · · (cid:18) − (cid:19)
12 = c m ( − m − , (62) follows. (cid:3) We can now proceed to the proof of our first main result.
Proof of Theorem A.
First, let λ = (2 m, , . . . , , ± , , . . . , ∈ Π k . Since ⊥ = id, Lemma 4.1and Lemma 4.3 together imply T n − k ◦ R k,n − k | H λ = c k,m ( − k ⊥ ∗ | H λ , (63)for some c k,m > φ ∈ P k and consider the L -orthogonal decomposition f φ = X λ ∈ Π k f ( λ ) φ , f ( λ ) φ ∈ H λ . (64)We shall use Lemma 2.12 for ϕ = F ( φ ) and ψ = φ · µ n − k . Recall also that µ n − k = c k µ n − k ,for some c k >
0, according to Theorem 2.2 (d). All in all, we have Q ( φ, φ ) = φ · φ · µ n − k = c k φ · ( φ · µ n − k ) (45) = c k Z Gr n − k f F ( φ ) ( E ) Kl φ · µ n − k ( E ) dE (44) = ˜ c k Z Gr n − k f F ( φ ) ( E ) [ T n − k ◦ R k,n − k ( f φ )] ( E ) dE (43) = ˜ c k Z Gr n − k f φ ( E ⊥ ) [ T n − k ◦ R k,n − k ( f φ )] ( E ) dE (64) = ˜ c k X λ ′ ,λ ∈ Π k Z Gr n − k f ( λ ′ ) φ ( E ⊥ ) h T n − k ◦ R k,n − k (cid:16) f ( λ ) φ (cid:17)i ( E ) dE (63) = ( − k X λ ′ ,λ ∈ Π k c k,λ Z Gr n − k f ( λ ′ ) φ ( E ⊥ ) f ( λ ) φ ( E ⊥ ) dE = ( − k X λ ′ ,λ ∈ Π k c k,λ Z Gr k f ( λ ′ ) φ ( F ) f ( λ ) φ ( F ) dF = ( − k X λ ∈ Π k c k,λ Z Gr n − k (cid:12)(cid:12)(cid:12) f ( λ ) φ ( F ⊥ ) (cid:12)(cid:12)(cid:12) dF, for some c k , ˜ c k , c k,λ >
0. Since f φ = 0, there is λ ∈ Π k with f ( λ ) φ = 0, and consequently,( − k Q ( φ, φ ) > , as desired. (cid:3) Hodge-Riemann relations for 1-homogeneous valuations
The purpose of this section is to prove the Hodge-Riemann relation for smooth valuationsof degree 1, regardless of parity. Using a different, yet analogous argumentation as in the evencase considered in §
4, the proof is based on a functional representation of these valuations, inparticular, on Alesker’s characterization theorem (Lemma 2.14) and recent results on sphericalvaluations due to Bernig and Hug (Lemma 2.13). Recall that the notation is kept from § Proposition 5.1.
Let ≤ k ≤ n . There is c k > such that for any f ∈ C ∞ ( S n − ) , one has µ n − ∗ µ k,f = c k µ k − ,f . (65) roof. Using the formulas (22) and (47), respectively, we indeed have µ n − ∗ µ k,f = 12 Z S n − f ( y ) ddλ (cid:12)(cid:12)(cid:12)(cid:12) λ =0 dS k ( K + λD, y )= c k Z S n − f ( y ) dS k − ( K, y )= c k µ k − ,f for some constant c k > (cid:3) Corollary 5.2.
Let ≤ k ≤ n − and q ∈ N with q = 1 . There is c k,q > such that for any f ∈ S q , one has µ · µ k,f = c k,q µ k +1 ,f . (66) Proof.
According to the previous proposition and Theorem 2.4, we have µ · µ k,f = F − ( F µ ∗ F µ k,f ) (50) = ˜ c k,q ( √− q F − ( µ n − ∗ µ n − k,f ) (65) = ˆ c k,q ( √− q F − µ n − k − ,f (50) = c k,q µ k +1 ,f , for some constants ˜ c k,q , ˆ c k,q , c k,q > (cid:3) We are now in position to prove the second main result of this article.
Proof of Theorem B.
Take an arbitrary non-zero φ ∈ P s . First, according to Lemma 2.14, φ = µ ,f for some f = P q f ( q ) , where f ( q ) ∈ S q , and the sum extends over all q ∈ N with q ≡ s mod 2 and q = 1. That φ is primitive means that0 = µ ,f · µ n − = X q µ ,f ( q ) · µ n − , = µ ,f (0) · µ n − , = c Z S n − f (0) ( y ) dy, for some c >
0. Since f (0) ∈ S is constant, this implies f (0) = 0. So we in fact have f = P q f ( q ) with q ≡ s mod 2 and q ≥
2. Then, Q ( φ, φ ) = µ ,f · µ ,f · µ n − = X q,r µ ,f ( q ) · µ ,f ( r ) · µ n − = X q,r c r µ ,f ( q ) · µ n − ,f ( r ) (51) = X q ˜ c q ( − q (cid:18) − q ( n + q − n − (cid:19) Z S n − (cid:12)(cid:12)(cid:12) f ( q ) ( y ) (cid:12)(cid:12)(cid:12) dy, for some c r , ˜ c q >
0. Observe that ( − q = ( − s and1 − q ( n + q − n − − q )( n − q ) n − q, n ≥
2. Finally, since φ = 0, there is q with f ( q ) = 0, and therefore( − s Q ( φ, φ ) < . (cid:3) emark 5.3. Let us point out that the method of the current section is, alas, not sufficient toobtain even partial results in higher degrees as all spherical valuations then obviously lie in theimage of the Lefschetz map, i.e., they are not primitive unless trivial.6.
Hodge-Riemann relations in the language of the Bernig-Fu convolution
We now reformulate both the proved and the conjectured Hodge-Riemann relations in termsof the other canonical multiplicative structure on Val ∞ . As we shall see, the setting of theBernig-Fu convolution is more suitable for applications to isoperimetric inequalities, and leadsthe way to further generalizations.Let 0 ≤ k ≤ ⌊ n ⌋ . We say that a valuation φ ∈ Val ∞ n − k of co-degree k is co-primitive if φ ∗ µ ∗ ( n − k +1) n − = 0 , (67)and denote the subspace of all such by C n − k . We also write C sn − k = C n − k ∩ Val s, ∞ , s = 0 , sn − k = F P sk . (68)In analogy with (56) and in agreement with the introduction, we define the Bernig-Fu-Hodge-Riemann pairing e Q : Val ∞ n − k × Val ∞ n − k → C by e Q ( φ, ψ ) = φ ∗ ψ ∗ µ ∗ ( n − k ) n − . (69)The key to understand the (perhaps unexpected) dependence of (4) on the parity of a valu-ation turns out to be the following observation, whose proof is due to S. Alesker: Lemma 6.1.
Let ≤ k ≤ n and let φ ∈ Re Val ∞ k be a real-valued valuation. (a) If φ is even, then F φ is real valued. (b) If φ is odd, then F φ is purely imaginary valued.Proof. (a) See [8], Theorem 5.4.1 (3).(b) Because F = − id on Val ∞ , , we may assume k ≤ ⌊ n ⌋ . First, the case k = 0 is trivial sinceVal ∞ , = { } . Second, for k = 1 the claim follows at once from (50). Finally, assume k ≥ φ ∈ Re Val , ∞ and ψ ∈ Re Val , ∞ k − . By what has been already shown, F φ is purelyimaginary while F ψ is real. Consequently, F ( φ · ψ ) = F φ ∗ F ψ is purely imaginary. Clearly, theirreducibility theorem (Theorem 2.1) holds verbatim for Re Val s, ∞ l , though for real subspaces.According to this fact, the real subspacespan n φ · ψ ; φ ∈ Re Val , ∞ , ψ ∈ Re Val , ∞ k − o ⊂ Re Val , ∞ k , which is GL( n, R ) invariant by (25), and obviously non-trivial (see Lemma 2.14 and Corollary2.8), is dense. Now the claim easily follows from linearity and continuity of F . (cid:3) Proposition 6.2.
For any φ ∈ Val ∞ k , one has e Q ( F φ, F φ ) = Q ( φ , φ ) − Q ( φ , φ ) , (70) where φ and φ are the even and odd part of φ , i.e., φ = φ + φ and φ s ∈ Val ∞ ,sk , s = 0 , .Proof. First, it is an immediate consequence of Lemma 6.1 thatRe F φ = F Re φ , Im F φ = F Im φ , while Re F φ = √− F Im φ , Im F φ = −√− F Re φ . sing this together with Theorem 2.4, one further deduces e Q ( F φ, F φ ) = e Q ( F φ , F φ ) + e Q ( F φ , F φ )= (cid:16) (Re F φ ) ∗ + (Im F φ ) ∗ + (Re F φ ) ∗ + (Im F φ ) ∗ (cid:17) ∗ µ ∗ ( n − k ) n − = (cid:16) ( F Re φ ) ∗ + ( F Im φ ) ∗ − ( F Im φ ) ∗ − ( F Re φ ) ∗ (cid:17) ∗ µ ∗ ( n − k ) n − = F − (cid:20) (cid:16) ( F Re φ ) ∗ + ( F Im φ ) ∗ − ( F Im φ ) ∗ − ( F Re φ ) ∗ (cid:17) ∗ µ ∗ ( n − k ) n − (cid:21) = (cid:16) (Re φ ) + (Im φ ) − (Im φ ) − (Re φ ) (cid:17) · µ n − k = Q ( φ , φ ) − Q ( φ , φ ) . (cid:3) Proposition 6.2 together with (68) implies at once that Conjecture C is indeed equivalentto Conjecture D. Along the same lines, an equivalent formulation of Theorems A and B,respectively, is the following:
Theorem 6.3.
Conjecture D is true under each of the following additional assumptions: (a) φ is even, (b) k = 1 . An isoperimetric-type inequality
In this section, an application of one of the proven cases of the Hodge-Riemann relations isdiscussed: It turns out that the purely algebraic statement of Theorem 6.3 (b) can be used todeduce an inequality of geometric type, namely, the first one of the isoperimetric inequalities(17). A special case (assuming n = 2 and K = − K ) of the argument we use was communicatedto us by S. Alesker. Corollary 7.1.
For any K ∈ K ∞ + , one has V ( K, D [ n − ≥ V ( K, K, D [ n − · vol n ( D ) . (71) Remark 7.2.
It follows at once from (16) that (71) is indeed equivalent to (cid:18) µ ( K ) µ ( D ) (cid:19) ≥ µ ( K ) µ ( D ) . (72)In another terminology, (71) is a special case of Minkowski’s second inequality (see [44], p. 382),and obviously a special case of the Aleksandrov-Fenchel inequality (18).
Proof.
Consider the following valuation: η = V ( · [ n − , K ) − V ( K, D [ n − n ( D ) V ( · [ n − , D ) . Clearly, η ∈ Val ∞ n − . Further, η ∈ C n − in fact since it follows from (16) and (21) that η ∗ µ ∗ ( n − n − = c η ∗ V ( · , D [ n − c (cid:20) V ( K, D [ n − − V ( K, D [ n − n ( D ) V ( D [ n ]) (cid:21) = 0 , for some c, ˜ c >
0. Consequently, according to Theorem 6.3 (b) and formulas (16) and (21),there are b, ˜ b, ˆ b > ≥ e Q ( η, η )= η ∗ η ∗ µ ∗ ( n − n − = b η ∗ η ∗ V ( · [2] , D [ n − b (cid:20) V ( · [ n − , K [2]) − V ( K, D [ n − n ( D ) V ( · [ n − , K, D ) V ( K, D [ n − vol n ( D ) V ( · [ n − , D [2]) ∗ V ( · [2] , D [ n − b " V ( K [2] , D [ n − − V ( K, D [ n − n ( D ) V ( K, D [ n − V ( K, D [ n − vol n ( D ) V ( D [ n ]) = ˆ b " V ( K [2] , D [ n − − V ( K, D [ n − vol n ( D ) , which is clearly equivalent to (71). (cid:3) Mixed Hodge-Riemann relations and the Aleksandrov–Fenchel inequality
We conclude by generalizing the arguments of the previous section to the conjectured mixedversion of the Hodge-Riemann relations. In particular, this allows us to show that ConjectureE yields the Aleksandrov-Fenchel inequality as a special case.Assume, at first, k = 0. It follows at once from (21) that in this case Conjecture E isequivalent to the statement that for any K , . . . , K n ∈ K ∞ + , one has V ( K , . . . , K n ) > , (73)which is a well-known, yet non-trivial fact (see [44], Theorems 5.1.7 and 5.1.8).More interesting, however, is the next case: Corollary 8.1.
Conjecture E for k = 1 implies the Aleksandrov-Fenchel inequality (18) forconvex bodies from the class K ∞ + . Remark 8.2.
The full generality of the Aleksandrov-Fenchel inequality, namely, the extensionof its validity to the class K , is then achieved by the standard limiting argument (cf. Aleksan-drov’s second proof [3]). Proof.
The same argumentation is valid here as that of the proof of Corollary 7.1. Namely, takeany K , . . . , K n ∈ K ∞ + and consider the following ‘mixed’ version of the valuation η examinedtherein: ξ = V ( · [ n − , K ) − V ( K , K , K , . . . , K n ) V ( K , K , K , . . . , K n ) V ( · [ n − , K ) ∈ Val ∞ n − . Recall that V ( K , K , K , . . . , K n ) >
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