A homogeneous decomposition theorem for valuations on convex functions
aa r X i v : . [ m a t h . M G ] A ug A HOMOGENEOUS DECOMPOSITION THEOREMFOR VALUATIONS ON CONVEX FUNCTIONS
ANDREA COLESANTI, MONIKA LUDWIG, AND FABIAN MUSSNIGA
BSTRACT . The existence of a homogeneous decomposition for continuous and epi-translation invariantvaluations on super-coercive functions is established. Continuous and epi-translation invariant valuationsthat are epi-homogeneous of degree n are classified. By duality, corresponding results are obtained forvaluations on finite-valued convex functions.2000 AMS subject classification: 52B45 (26B25, 52A21, 52A41)
1. I
NTRODUCTION
Given a space of real-valued functions X , we consider real-valued valuations on X , that is, functionals Z : X → R such that(1) Z( u ∨ v ) + Z( u ∧ v ) = Z( u ) + Z( v ) for every u, v ∈ X with u ∨ v and u ∧ v ∈ X , where ∨ and ∧ denote the point-wise maximum andminimum, respectively. For X , the space of indicator functions of convex bodies (that is, compactconvex sets) in R n , we obtain the classical notion of valuation on convex bodies. Here strong structureand classification theorems have been established over the last seventy years (see [1, 2, 6, 7, 19–21, 28]for some recent results and [22, 23, 36] for information on the classical theory). The aim of this articleis to obtain such results also in the functional setting. In particular, we will establish a homogeneousdecomposition result `a la McMullen [30].Valuations on function spaces have only recently started to attract attention. Classification resultswere obtained for L p and Sobolev spaces [24–27, 29, 38, 39], spaces of quasi-convex functions [12, 13],of Lipschitz functions [17], of definable functions [4] and on Banach lattices [37]. Spaces of convexfunctions play a special role because of their close connection to convex bodies. Here classificationresults were obtained for SL( n ) invariant and for monotone valuations in [8, 14, 15, 32–34] and theconnection to valuations on convex bodies was explored by Alesker [3]. While the theory of translationinvariant valuations is well developed for convex bodies, for convex functions the corresponding theorydid not exist till now. We introduce the notion of epi-translation invariance to build such a theory. Inparticular, we will show that on the space of super-coercive convex functions there is a homogeneousdecomposition for continuous and epi-translation invariant valuations and there exist non-trivial suchvaluations for each degree of epi-homogeneity while on the larger space of coercive convex functions allcontinuous and epi-translation invariant valuations are constant.The general space of (extended real-valued) convex functions on R n is defined asConv ( R n ) = { u : R n → R ∪ { + ∞} : u is convex and lower semicontinuous , u + ∞} . It is equipped with the topology induced by epi-convergence (see Section 2.2). Continuity of valuationsdefined on Conv ( R n ) , or on subsets of Conv ( R n ) , will be always with respect to this topology. The spaceConv ( R n ) is a standard space in convex analysis (see [35]) and important in many applications. As we Key words and phrases.
Convex function, valuation, homogeneous decomposition. will show, Conv ( R n ) is too large for our purposes. We will be mainly interested in two of its subspaces.The first is formed by coercive functions,Conv coe ( R n ) = (cid:26) u ∈ Conv ( R n ) : lim | x |→ + ∞ u ( x ) = + ∞ (cid:27) , where | x | is the Euclidean norm of x ∈ R n . The second is formed by super-coercive functions,Conv sc ( R n ) = (cid:26) u ∈ Conv ( R n ) : lim | x |→ + ∞ u ( x ) | x | = + ∞ (cid:27) . The space of super-coercive convex functions is related to another subspace of Conv ( R n ) , formed byconvex functions with finite values,Conv ( R n ; R ) = (cid:8) v ∈ Conv ( R n ) : v ( x ) < + ∞ for all x ∈ R n (cid:9) . Indeed, v ∈ Conv ( R n ; R ) if and only if its standard conjugate or Legendre transform v ∗ belongs toConv sc ( R n ) (see Section 1.3).1.1. One of the most important structural results for valuations on convex bodies is the existence of ahomogeneous decomposition for translation invariant valuations. It was conjectured by Hadwiger andestablished by McMullen [30] (see Section 2.1). Our first aim is to establish such a result for valuationson convex functions. We define epi-multiplication by setting for u ∈ Conv ( R n ) and λ > , λ u ( x ) = λ u (cid:16) xλ (cid:17) for x ∈ R n . From a geometric point of view, this operation has the following meaning: the epigraphof λ u is obtained by rescaling the epigraph of u by the factor λ . We extend the definition of epi-multiplication to u ( x ) = 0 if x = 0 and u ( x ) = + ∞ if x = 0 . It is easy to see that u ∈ Conv sc ( R n ) implies λ u ∈ Conv sc ( R n ) for λ ≥ . A functional Z :
Conv sc ( R n ) → R is called epi-homogeneous ofdegree α ∈ R if Z( λ u ) = λ α Z( u ) for all u ∈ Conv sc ( R n ) and λ > . Here and in the following corresponding definitions will be used forConv ( R n ) and its subspaces.We call Z :
Conv sc ( R n ) → R translation invariant if Z( u ◦ τ − ) = Z( u ) for every u ∈ Conv sc ( R n ) and every translation τ : R n → R n . If u ∈ Conv sc ( R n ) then u ◦ τ − ∈ Conv sc ( R n ) as well. We say that Z is vertically translation invariant if Z( u + α ) = Z( u ) for all u ∈ Conv sc ( R n ) and α ∈ R . If Z is both translation invariant and vertically translation invariant,then Z is called epi-translation invariant . As we will see, the set of continuous, epi-translation invariantvaluations on Conv sc ( R n ) is non-empty. Note that a functional Z is epi-translation invariant if for all u ∈ Conv sc ( R n ) the value Z( u ) is not changed by translations of the epigraph of u .The following result establishes a homogeneous decomposition for continuous and epi-translationinvariant valuations on Conv sc ( R n ) . Theorem 1. If Z :
Conv sc ( R n ) → R is a continuous and epi-translation invariant valuation, then thereare continuous and epi-translation invariant valuations Z , . . . , Z n : Conv sc ( R n ) → R such that Z i isepi-homogeneous of degree i and Z = Z + · · · + Z n . We will see that this theorem is no longer true if we remove the condition of vertical translationinvariance (see Section 8). We will also see that the set of continuous and epi-translation invariantvaluations is trivial on the larger set of coercive convex functions (see Section 9). Hence the assumptionof super-coercivity is in some sense necessary.
Milman and Rotem [31] discuss the problem to find a functional analog of Minkowski’s mixed volumetheorem. In particular, they point out that such a result is not possible on Conv ( R n ) for inf-convolutionas addition and the volume functional u R R n e − u ( x ) d x . Instead, they define a new addition for convexfunctions to obtain a functional mixed volume theorem. A consequence of Theorem 1 is that continuousand epi-translation invariant valuations are multilinear on Conv sc ( R n ) with respect to inf-convolution andepi-multiplication (see Theorem 21). Thus, for all such valuations, a functional analog of Minkowski’smixed volume theorem is obtained on Conv sc ( R n ) with inf-convolution as addition.1.2. The following result gives a characterization of continuous and epi-translation invariant valuationson Conv sc ( R n ) , which are epi-homogeneous of degree n . For u ∈ Conv sc ( R n ) , we denote by dom( u ) the set of points of R n where u is finite and by ∇ u the gradient of u . Note that by standard propertiesof convex functions, ∇ u ( x ) is well defined for a.e. x ∈ dom( u ) . Let C c ( R n ) be the set of continuousfunctions with compact support on R n . Theorem 2.
A functional
Z :
Conv sc ( R n ) → R is a continuous and epi-translation invariant valuationthat is epi-homogeneous of degree n , if and only if there exists ζ ∈ C c ( R n ) such that Z( u ) = Z dom( u ) ζ ( ∇ u ( x )) d x for every u ∈ Conv sc ( R n ) . We will also obtain a classification of continuous and epi-translation invariant valuations that are epi-homogeneous of degree . These are just constants. As a consequence of these results and Theorem 1,we obtain the following complete classification in dimension one. Corollary 3.
A functional
Z :
Conv sc ( R ) → R is a continuous and epi-translation invariant valuation,if and only if there exist a constant ζ ∈ R and a function ζ ∈ C c ( R ) such that Z( u ) = ζ + Z dom( u ) ζ ( u ′ ( x )) d x for every u ∈ Conv sc ( R ) . ( R n ; R ) and Conv sc ( R n ) given by thestandard conjugate, or Legendre transform, of convex functions. For u ∈ Conv ( R n ) , we denote by u ∗ itsconjugate, defined by u ∗ ( y ) = sup x ∈ R n ( h x, y i − u ( x )) for y ∈ R n , where h x, y i is the inner product of x, y ∈ R n . Note that u ∈ Conv sc ( R n ) if and only if u ∗ ∈ Conv ( R n ; R ) (see, for example, [35, Theorem 11.8]).Let Z be a continuous valuation on Conv ( R n ; R ) . It was proved in [16] that Z ∗ : Conv sc ( R n ) → R ,defined by Z ∗ ( u ) = Z( u ∗ ) , is a continuous valuation as well. This fact permits to transfer results for valuations on Conv ( R n ; R ) toresults valid for valuations on Conv sc ( R n ) and vice versa. We call Z ∗ the dual valuation of Z .A valuation Z on Conv ( R n ; R ) is called homogeneous if there exists α ∈ R such that Z( λv ) = λ α Z( v ) for all v ∈ Conv ( R n ; R ) and λ ≥ . We say that Z is dually translation invariant if for every linearfunction ℓ : R n → R Z( v + ℓ ) = Z( v ) for every v ∈ Conv ( R n ; R ) . Let ℓ ( y ) = h y, x i for x , y ∈ R n . As ( v + ℓ ) ∗ ( x ) = v ∗ ( x − x ) for v ∈ Conv ( R n ; R ) , we see that Z is dually translation invariant if and only if Z ∗ is translation invariant. We ANDREA COLESANTI, MONIKA LUDWIG, AND FABIAN MUSSNIG define vertical translation invariance for valuations on Conv ( R n ; R ) in the same way as on Conv sc ( R n ) .We say that Z is dually epi-translation invariant on Conv ( R n ; R ) if it is vertically and dually translationinvariant. Note that a functional Z is dually epi-translation invariant, if for all v ∈ Conv ( R n ; R ) , thevalue Z( v ) is not changed by adding an affine function to v .Let Z be a valuation on Conv ( R n ; R ) . We note the following simple facts. The valuation Z is verticallytranslation invariant if and only if Z ∗ has the same property. The valuation Z ∗ is epi-homogeneous ofdegree α if and only if Z is homogeneous of degree α .Hence we obtain the following result as a consequence of Theorem 1. Theorem 4. If Z :
Conv ( R n ; R ) → R is a continuous and dually epi-translation invariant valuation,then there are continuous and dually epi-translation invariant valuations Z , . . . , Z n : Conv ( R n ; R ) → R such that Z i is homogeneous of degree i and Z = Z + · · · + Z n . Alesker [3] introduced the following class of valuations on Conv ( R n ; R ) . Given real symmetric n × n matrices M , . . . , M n , denote by det( M , . . . , M n ) their mixed discriminant. Let i ∈ { , . . . , n } andwrite det( M [ i ] , M , . . . , M n − i ) for the mixed discriminant in which the matrix M is repeated i times.Let A , . . . , A n − i be continuous, symmetric n × n matrix-valued functions on R n with compact supportand ζ ∈ C c ( R n ) . Given a function v ∈ Conv ( R n ; R ) ∩ C ( R n ) , set(2) Z( v ) = Z R n ζ ( x ) det(D v ( x )[ i ] , A ( x ) , . . . , A n − i ( x )) d x where D v is the Hessian matrix of v . Alesker [3] proved that Z can be extended to a continuous valuationon Conv ( R n ; R ) . Valuations of type (2) are homogeneous of degree i and dually epi-translation invariant.This implies in particular that the set of valuations with these properties is non-empty. Clearly, the dualfunctional Z ∗ is a continuous, epi-translation invariant, epi-homogeneous valuation on Conv sc ( R n ) .Next, we state the counterpart of Theorem 2 for valuations on Conv ( R n ; R ) . Let Θ ( v, · ) be theHessian measure of order of a function v ∈ Conv ( R n ; R ) (see Section 4 for the definition). Theorem 5.
A functional
Z :
Conv ( R n ; R ) → R is a continuous and dually epi-translation invariantvaluation that is homogeneous of degree n , if and only if there exists ζ ∈ C c ( R n ) such that Z( v ) = Z R n × R n ζ ( x ) dΘ ( v, ( x, y )) for every v ∈ Conv ( R n ; R ) . In the special case of dimension one, we obtain the following complete classification theorem.
Corollary 6.
A functional
Z :
Conv ( R ; R ) → R is a continuous and dually epi-translation invariantvaluation, if and only if there exist a constant ζ ∈ R and a function ζ ∈ C c ( R ) such that Z( v ) = ζ + Z R × R ζ ( x ) dΘ ( v, ( x, y )) for every v ∈ Conv ( R ; R ) . The plan for this paper is as follows. In Section 2, we collect results on convex bodies and functionsneeded for the proofs of the main results. In Section 3, an inclusion-exclusion principle is establishedfor valuations on convex functions and in Section 4, the existence and properties of the valuations inTheorem 2 and Theorem 5 are deduced by using results on Hessian valuations. Theorem 1 is proved inSection 5. As a consequence the polynomiality of epi-translation invariant valuations is obtained and aconnection to the valuations introduced by Alesker is established in Section 6. The proof of Theorem 2 isgiven in Section 7. In the final sections, the necessity of the assumptions in Theorem 1 is demonstrated.
2. P
RELIMINARIES
We work in n -dimensional Euclidean space R n , with n ≥ , endowed with the Euclidean norm | · | and the usual scalar product h· , ·i .2.1. A convex body is a nonempty, compact and convex subset of R n . The family of all convex bodiesis denoted by K n . A polytope is the convex hull of finitely many points in R n . The set of polytopes,denoted by P n , is contained in K n . We equip both K n and P n with the topology coming from theHausdorff metric.A functional Z : K n → R is a valuation if Z( K ∪ L ) + Z( K ∩ L ) = Z( K ) + Z( L ) for every K, L ∈ K n with K ∪ L ∈ K n . We say that Z is translation invariant if Z( τ K ) = Z( K ) for alltranslations τ : R n → R n and K ∈ K n . It is homogeneous of degree α ∈ R , if Z( λ K ) = λ α Z( K ) forall K ∈ K n and λ ≥ .The following result by McMullen [30] establishes a homogeneous decomposition for continuous andtranslation invariant valuations on K n . Theorem 7 (McMullen) . If Z : K n → R is a continuous and translation invariant valuation, then thereare continuous and translation invariant valuations Z , . . . , Z n : K n → R such that Z i is homogeneousof degree i and Z = Z + · · · + Z n . We recall two classification results for valuations on convex bodies. First, we note that it is easy tosee that every continuous and translation invariant valuation that is homogeneous of degree is constant.The classification of continuous and translation invariant valuations that are n -homogeneous is due toHadwiger [22]. Let V n denote n -dimensional volume (that is, n -dimensional Lebesgue measure). Theorem 8 (Hadwiger) . A functional
Z : K n → R is a continuous and translation invariant valuationthat is homogeneous of degree n , if and only if there exists α ∈ R such that Z = α V n . A ⊂ R n , let I A : R n → R ∪ { + ∞} denote the (convex) indicatrix function of A , I A ( x ) = (cid:26) if x ∈ A , + ∞ if x / ∈ A .Note that for a convex body K , we have I K ∈ Conv sc ( R n ) .We equip Conv ( R n ) with the topology associated to epi-convergence. Here a sequence u k ∈ Conv ( R n ) is epi-convergent to u ∈ Conv ( R n ) if for all x ∈ R n the following conditions hold:(i) For every sequence x k that converges to x , we have u ( x ) ≤ lim inf k →∞ u k ( x k ) .(ii) There exists a sequence x k that converges to x such that u ( x ) = lim k →∞ u k ( x k ) .The following result can be found in [35, Theorem 11.34]. Proposition 9.
A sequence u k of functions from Conv ( R n ) epi-converges to u ∈ Conv ( R n ) if and only ifthe sequence u ∗ k epi-converges to u ∗ . If u ∈ Conv coe ( R n ) , then for t ∈ R the sublevel sets { u ≤ t } = { x ∈ R n : u ( x ) ≤ t } are eitherempty or in K n . The next result, which follows from [15, Lemma 5] and [5, Theorem 3.1], shows thaton Conv coe ( R n ) epi-convergence is equivalent to Hausdorff convergence of sublevel sets, where we saythat { u k ≤ t } → ∅ as k → ∞ if there exists k ∈ N such that { u k ≤ t } = ∅ for k ≥ k . Lemma 10. If u k , u ∈ Conv coe ( R n ) , then u k epi-converges to u if and only if { u k ≤ t } → { u ≤ t } forevery t ∈ R with t = min x ∈ R n u ( x ) . ANDREA COLESANTI, MONIKA LUDWIG, AND FABIAN MUSSNIG v ∈ Conv ( R n ; R ) is called piecewise affine if there exist finitely many affine functions w , . . . , w m : R n → R such that(3) v = m _ i =1 w i . The set of piecewise affine functions will be denoted by
Conv p . a . ( R n ; R ) . It is a subset of Conv ( R n ; R ) .We recall that epi-convergence in Conv ( R n ; R ) is equivalent to uniform convergence on compactsets (see, for example, [35, Theorem 7.17]). Hence the following proposition follows from standardapproximation results for convex functions. Proposition 11.
For every v ∈ Conv ( R n ; R ) , there exists a sequence in Conv p . a . ( R n ; R ) which epi-converges to v . We also need to introduce a counterpart of
Conv p . a . ( R n ; R ) in Conv sc ( R n ) . For given polytopes P, P , . . . , P m ∈ P n , the collection { P , . . . , P m } is called a polytopal partition of P if P = S mi =1 P i andthe P i ’s have pairwise disjoint interiors. A function u ∈ Conv sc ( R n ) belongs to Conv p . a . ( R n ) if thereexists a polytope P and a polytopal partition { P , . . . , P m } of P such that u = m ^ i =1 ( w i + I P i ) where w , . . . , w m : R n → R are affine.By [35, Theorem 11.14], a function u is in Conv p . a . ( R n ) if and only if u ∗ is in Conv p . a . ( R n ; R ) . Hence,we obtain the following consequence of Proposition 9 and Proposition 11. Corollary 12.
For every u ∈ Conv sc ( R n ) , there exists a sequence in Conv p . a . ( R n ) which epi-convergesto u . Since Conv sc ( R n ) is a dense subset of Conv coe ( R n ) , it is easy to see that the statement of Corollary 12also holds if Conv sc ( R n ) is replaced by Conv coe ( R n ) .3. T HE I NCLUSION -E XCLUSION P RINCIPLE
It is often useful to extend the valuation property (1) to several convex functions. For valuations onconvex bodies, this is an important tool and a consequence of Groemer’s extension theorem [18]. For m ≥ and u , . . . , u m ∈ Conv ( R n ) , we set u J = W j ∈ J u j for ∅ 6 = J ⊂ { , . . . , m } . Let | J | denote thenumber of elements in J . Theorem 13. If Z :
Conv ( R n ) → R is a continuous valuation, then (4) Z( u ∧ · · · ∧ u m ) = X ∅6 = J ⊂{ ,...,m } ( − | J |− Z( u J ) for all u , . . . , u m ∈ Conv ( R n ) and m ∈ N whenever u ∧ · · · ∧ u m ∈ Conv ( R n ) . Note that Conv coe ( R n ) and Conv sc ( R n ) are closed under the operation of taking maxima. HenceTheorem 13 also holds with Conv ( R n ) replaced by one of these spaces. Let V Conv ( R n ) denote the set of finite minima of convex functions from Conv ( R n ) . It is easy to seethat V Conv ( R n ) is a lattice. If Z is a valuation on a lattice, a simple induction argument shows that theinclusion-exclusion principle (4) holds. Hence Theorem 13 is a consequence of the following extensionresult. Theorem 14.
A continuous valuation on
Conv ( R n ) admits a unique extension to a valuation on thelattice V Conv ( R n ) . We identify a convex function with its epigraph. Let C n +1 epi be the set of closed convex sets in R n +1 that are epigraphs of functions in Conv ( R n ) and equip this set with the Painlev´e-Kuratowski topology,which corresponds to the topology induced by epi-convergence (see, for example, [35, Definition 7.1]).A slight modification of Groemer’s extension theorem [18] (or see [36, Theorem 6.2.3] or [23]) showsthat the following statement is true (we omit the proof). Here S C n +1 epi is the set of all finite unions ofelements from C n +1 epi . Theorem 14 is equivalent to Theorem 15. Theorem 15.
A continuous valuation on C n +1 epi admits a unique extension to a valuation on the lattice S C n +1 epi . We require the following simple consequence of the inclusion-exclusion principle, Theorem 13 and ofCorollary 12.
Lemma 16.
Let Z be a continuous valuation on Conv sc ( R n ) (or on Conv coe ( R n ) ). If (5) Z( w + I P ) = 0 for every affine function w : R n → R and for every polytope P , then Z ≡ .Proof. By Corollary 12 (and the remark following it), it suffices to prove that Z( u ) = 0 for u ∈ Conv sc ( R n ) (or u ∈ Conv coe ( R n ) ) that is piecewise affine. So, let u = V mi =1 ( w i + I P i ) with w , . . . , w m affine and P , . . . , P m ∈ P n . By Theorem 13 (and the remark following it), it is enough toshow that Z _ j ∈ J ( w j + I P j ) ! = 0 for every ∅ 6 = J ⊂ { , . . . , m } . This follows from (5) as W j ∈ J ( w j + I P j ) is a piecewise affine functionrestricted to a polytope. (cid:3)
4. H
ESSIAN MEASURES AND VALUATIONS
For u ∈ Conv ( R n ) and x ∈ R n , we denote by ∂u ( x ) the subgradient of u at x , that is, ∂u ( x ) = { y ∈ R n : u ( z ) ≥ u ( x ) + h z − x, y i for all z ∈ R n } . We set Γ u = { ( x, y ) ∈ R n × R n : y ∈ ∂u ( x ) } . In other words, Γ u is the generalized graph of ∂u .Next, we recall the notion of Hessian measures of a function u ∈ Conv ( R n ) . These are non-negativeBorel measures defined on the Borel subsets of R n × R n , which we will denote by Θ i ( u, · ) with i =0 , . . . , n . Their definition can be given as follows (see also [10, 11, 16]). Let η ⊂ R n × R n be a Borel setand s > . Consider the following set P s ( u, η ) = { x + sy : ( x, y ) ∈ Γ u ∩ η } . ANDREA COLESANTI, MONIKA LUDWIG, AND FABIAN MUSSNIG
It can be proven (see Theorem 7.1 in [16]) that P s ( u, η ) is measurable and that its measure is a polynomialin the variable s , that is, there exists ( n + 1) non-negative coefficients Θ i ( u, η ) such that H n ( P s ( u, η )) = n X i =0 (cid:18) ni (cid:19) s i Θ n − i ( u, η ) . Here H n is the n -dimensional Hausdorff measure in R n , normalized so that it coincides with the Lebesguemeasure in R n . The previous formula defines the Hessian measures of u ; for more details we refer thereader to [10, 11, 16].According to Theorem 8.2 in [16], for every v ∈ Conv ( R n ; R ) and for every Borel subset η of R n × R n (6) Θ i ( v, η ) = Θ n − i ( v ∗ , ˆ η ) , where ˆ η = { ( x, y ) ∈ R n × R n : ( y, x ) ∈ η } .We require the following statement for Hessian valuations for i = 0 . As the proof is the same for allindices i , we give the more general statement. Let [D v ( x )] i be the i -th elementary symmetric functionof the eigenvalues of the Hessian matrix D v . Theorem 17.
Let ζ ∈ C ( R × R n × R n ) have compact support with respect to the second variable. For i ∈ { , , . . . , n } , (7) Z( v ) = Z R n × R n ζ ( v ( x ) , x, y ) dΘ i ( v, ( x, y )) is well defined for every v ∈ Conv ( R n ; R ) and defines a continuous valuation on Conv ( R n ; R ) . More-over, (8) Z( v ) = Z R n ζ ( v ( x ) , x, ∇ v ( x )) [D v ( x )] n − i d x for every v ∈ Conv ( R n ; R ) ∩ C ( R n ) . We use the following result.
Theorem 18 ([16], Theorem 1.1) . Let ζ ∈ C ( R × R n × R n ) have compact support with respect to thesecond and third variables. For every i ∈ { , , . . . , n } , the functional defined by v Z R n × R n ζ ( v ( x ) , x, y ) dΘ i ( v, ( x, y )) defines a continuous valuation on Conv ( R n ) . Moreover, (9) Z( v ) = Z R n ζ ( v ( x ) , x, ∇ v ( x )) [D v ( x )] n − i d x for v ∈ Conv ( R n ) ∩ C ( R n ) .Proof of Theorem 17. Since ζ has compact support with respect to the second variable, there is r > such that ζ ( t, x, y ) = 0 for every y ∈ R n with | y | ≥ r and ( t, x ) ∈ R × R n . Let v, v k ∈ Conv ( R n ; R ) be such that v k epi-converges to v . Since the functions are convex and finite this implies uniform con-vergence on compact sets, in particular on B r := { x ∈ R n : | x | ≤ r } . Moreover, the sequence v k isuniformly bounded on B r and uniformly Lipschitz. Hence, there exists c > such that | v k ( x ) | ≤ c, | v ( x ) | ≤ c, | y | ≤ c for all k ∈ N , x ∈ B r and y ∈ ∂v k ( x ) ∪ ∂v ( x ) . Next, let η : R n → R be smooth with compact support such that η ( y ) = 1 for all y ∈ R n with | y | ≤ c and define ˜ ζ ∈ C ( R × R n × R n ) by ˜ ζ ( t, x, y ) = ζ ( t, x, y ) η ( y ) . The function ˜ ζ satisfies the conditions of Theorem 18 and ζ ( v ( x ) , x, y ) = ˜ ζ ( v ( x ) , x, y ) for all x ∈ R n , y ∈ ∂v ( x ) and ζ ( v k ( x ) , x, y ) = ˜ ζ ( v k ( x ) , x, y ) for all x ∈ R n , y ∈ ∂v k ( x ) and k ∈ N . Hence, byTheorem 18, Z R n × R n ζ ( v k ( x ) , x, y ) dΘ i ( v k , ( x, y )) = Z R n × R n ˜ ζ ( v k ( x ) , x, y ) dΘ i ( v k , ( x, y )) −→ Z R n × R n ˜ ζ ( v ( x ) , x, y ) dΘ i ( v, ( x, y )) = Z R n × R n ζ ( v ( x ) , x, y ) dΘ i ( v, ( x, y )) as k → ∞ . Since v and v k were arbitrary this shows that (7) is well defined and continuous. Since sucha function ˜ ζ can especially be found for any finite number of functions in Conv ( R n ; R ) , this also provesthe valuation property. Property (8) follows from (9). (cid:3) As a simple consequence of Theorem 17 we obtain the following statement.
Proposition 19.
For ζ ∈ C c ( R n ) , the functional Z :
Conv ( R n ; R ) → R , defined by (10) Z( v ) = Z R n × R n ζ ( x ) dΘ ( v, ( x, y )) , is a continuous, dually epi-translation invariant valuation which is is homogeneous of degree n .Proof. By Theorem 17 the map defined by (10) is a continuous valuation on on Conv ( R n ; R ) . It remainsto show dually epi-translation invariance. For v ∈ Conv ( R n ; R ) ∩ C ( R n ) it follows from (8) that Z( v ) = Z R n ζ ( x ) det(D v ( x )) d x which is clearly invariant under the addition of constants and linear terms. The statement now easilyfollows for general v ∈ Conv ( R n ; R ) by approximation. (cid:3) By the considerations presented in Section 1.3, (6) and Proposition 19 lead to the following result.
Proposition 20.
For ζ ∈ C c ( R n ) , the functional Z :
Conv sc ( R n ) → R , defined by Z( u ) = Z R n × R n ζ ( y ) dΘ n ( u, ( x, y )) , is a continuous and epi-translation invariant valuation on Conv sc ( R n ) which is epi-homogeneous ofdegree n . Note, that if Z is as in Proposition 20, then Z( u ) = Z R n × R n ζ ( y ) dΘ n ( u, ( x, y )) = Z dom( u ) ζ ( ∇ u ( x )) d x for every u ∈ Conv sc ( R n ) . See also [16, Section 10.4].
5. P
ROOF OF T HEOREM y ∈ R n , define the linear function ℓ y : R n → R as ℓ y ( x ) = h x, y i . For K ∈ K n , the function ℓ y + I K belongs to Conv sc ( R n ) . Claim.
The functional ˜Z y : K n → R , defined by ˜Z y ( K ) = Z( ℓ y + I K ) , is a continuous and translation invariant valuation.Proof. i) The valuation property. Let
K, L ∈ K n be such that K ∪ L ∈ K n . Note that ( ℓ y + I K ) ∨ ( ℓ y + I L ) = ℓ y + I K ∩ L ; ( ℓ y + I K ) ∧ ( ℓ y + I L ) = ℓ y + I K ∪ L . Hence the valuation property of Z implies that ˜Z y is a valuation. ii) Translation invariance. Let x ∈ R n . For every x ∈ R n we have ℓ y ( x ) + I K + x ( x ) = h x, y i + I K ( x − x )= h x − x , y i + I K ( x − x ) + h x , y i = ℓ y ( x − x ) + I K ( x − x ) + h x , y i . In other words, the functions ℓ y + I K + x and ℓ y + I K differ only by a translation of the variable and byan additive constant. Using the epi-translation invariance of Z we get ˜Z y ( K + x ) = Z( ℓ y + I K + x ) = Z( ℓ y + I K ) = ˜Z y ( K ) . iii) Continuity. By Lemma 10, a sequence of convex bodies K i converges to K if and only if ℓ y + I K i epi-converges to ℓ y + I K . Hence the continuity of Z implies that of ˜Z y . (cid:3) Let y ∈ R n be fixed. By the previous claim and Theorem 7, there exist continuous and translationinvariant valuations ˜Z y, , . . . , ˜Z y,n on K n such that ˜Z y,j is j -homogeneous and ˜Z y = n X j =0 ˜Z y,j . Let K ∈ K n . For λ ≥ , we have λ ( ℓ y + I K ) = ℓ y + I λK . Therefore we obtain, for every λ ≥ , Z( λ ( ℓ y + I K )) = n X j =0 ˜Z y,j ( K ) λ j . We consider the system of equations,(11) Z( k ( ℓ y + I K )) = n X j =0 ˜Z y,j ( K ) k j , k = 0 , , . . . , n. Its associated matrix is a Vandermonde matrix and invertible. Hence there are coefficients α ij for i, j =0 , . . . n , such that ˜Z y,i ( K ) = n X j =0 α ij Z( k ( ℓ y + I K )) , i = 0 , . . . , n. Note that the coefficients α ij are independent of y and K . For i = 0 , . . . , n , we define Z i : Conv sc ( R n ) → R as Z i ( u ) = n X j =0 α ij Z( j u ) . In general, if Z is a continuous, epi-translation invariant valuation on Conv sc ( R n ) and λ ≥ , then thefunctional u Z( λ u ) is a continuous, epi-translation valuation as well. Hence Z i is a continuous,epi-translation invariant valuation on Conv sc ( R n ) , for every i = 0 , . . . , n .By (11) and the definition of Z i , for every y ∈ R n and K ∈ K n we may write Z i ( ℓ y + I K ) = ˜Z y,i ( K ) . Therefore Z( ℓ y + I K ) = n X i =0 Z i ( ℓ y + I K ) . Moreover, by the homogeneity of the Z y,i we have, for λ ≥ , Z i ( λ ( ℓ y + I K )) = ˜Z y,i ( λK ) = λ i ˜Z y,i ( K ) = λ i Z i ( ℓ y + I K ) . As a conclusion, we have the following statement: there exist continuous and epi-translation invariantvaluations Z , . . . , Z n on Conv sc ( R n ) such that, for every y ∈ R n and for every K ∈ K n , setting u = ℓ y + I K , we have Z( u ) = n X i =0 Z i ( u ) , and, for every λ ≥ , Z i ( λ u ) = λ i Z i ( u ) . The same statement holds if we replace u = ℓ y + I K by u = ℓ y + I K + α , for any constant α ∈ R as allvaluations involved are vertically translation invariant.If we apply Lemma 16 to Z − n X i =0 Z i , we get that this valuation vanishes on Conv sc ( R n ) , so that Z( u ) = n X i =0 Z i ( u ) for every u ∈ Conv sc ( R n ) . For λ ≥ , the same lemma applied to the valuation on Conv sc ( R n ) definedby u Z i ( λ u ) − λ i Z i ( u ) , shows that this must be identically zero as well, that is, Z i is epi-homogeneous of degree i . The proof iscomplete.
6. P
OLYNOMIALITY
In this section we establish the polynomial behavior of continuous and epi-translation invariant val-uations on Conv sc ( R n ) . This corresponds to the polynomiality of translation invariant valuations onconvex bodies stated by Hadwiger and proved by McMullen [30]. We start by recalling the defini-tion of inf-convolution (see, for example, [35, 36]). For u, v ∈ Conv ( R n ) , we define the function u (cid:3) v : R n → [ −∞ , + ∞ ] by u (cid:3) v ( z ) = inf { u ( x ) + v ( y ) : x, y ∈ R n , x + y = z } for z ∈ R n . This operation can be extended to more than two functions with corresponding coefficients.The inf-convolution has a straightforward geometric meaning: the epigraph of u (cid:3) v is the Minkowskisum of the epigraphs of u and v .By [36, Section 1.6], for every α, β > and for every u, v ∈ Conv ( R n ) , we have α u (cid:3) β u ∈ Conv ( R n ) , if this function does not attain −∞ . Moreover, in this case we have thefollowing relation (see for instance [9, Proposition 2.1]): ( α u (cid:3) β v ) ∗ = ( αu ∗ + βv ∗ ) . This shows in particular that if u, v ∈ Conv sc ( R n ) then α u (cid:3) β v ∈ Conv sc ( R n ) . Indeed, in this case u ∗ and v ∗ belong to Conv ( R n ; R ) and so does their usual sum. Consequently, its conjugate belongs toConv sc ( R n ) . We say that Z is epi-additive if Z( α u (cid:3) β v ) = α Z( u ) + β Z( v ) for all α, β > and u, v ∈ Conv sc ( R n ) .Let Z :
Conv sc ( R n ) → R be a continuous, epi-translation invariant valuation that is epi-homogeneousof degree m ∈ { , . . . , n } . For u ∈ Conv sc ( R n ) , we consider the functional Z u : Conv sc ( R n ) → R defined by Z u ( u ) = Z( u (cid:3) u ) . The functional Z u is a continuous and epi-translation invariant valuation on Conv sc ( R n ) . Indeed, thevaluation property, continuity and vertical translation invariance follow immediately from the corre-sponding properties of Z . As for translation invariance, let x ∈ R n and τ : R n → R n be the translationby x , that is, τ ( x ) = x + x . We have ( u ◦ τ − ) (cid:3) u = (cid:0) ( u ◦ τ − ) ∗ + u ∗ (cid:1) ∗ = ( u ∗ + h· , x i + u ∗ ) ∗ = ( u (cid:3) u ) ◦ τ − . Hence the epi-translation invariance of Z u follows from the epi-translation invariance of Z . Therefore,we may apply Theorem 1 to obtain a polynomial expansion Z(( λ u ) (cid:3) u ) = Z u ( λ u ) = n X i =0 λ i Z u ,i ( u ) for λ ≥ and u ∈ Conv sc ( R n ) , where the functionals Z u ,i are continuous, epi-translation invariantvaluations on Conv sc ( R n ) that are epi-homogeneous of degree i ∈ { , . . . , n } .Similarly, for fixed ¯ u ∈ Conv sc ( R n ) one can show that v Z v,i (¯ u ) defines a continuous and epi-translation invariant valuation on Conv sc ( R n ) . Hence, as in the proof of Theorem 6.3.4 in [36], we mayrepeat this argument to obtain the following statement. Theorem 21.
Let
Z :
Conv sc ( R n ) → R be a continuous and epi-translation invariant valuation that isepi-homogeneous of degree m ∈ { , . . . , n } . There exists a symmetric function ¯Z : ( Conv sc ( R n )) m → R such that for k ∈ N , u , . . . , u k ∈ Conv sc ( R n ) and λ , . . . , λ k ≥ , Z( λ u (cid:3) · · · (cid:3) λ k u k ) = X i ,...,i k ∈{ ,...,m } i + ··· + i k = m (cid:18) mi · · · i k (cid:19) λ i · · · λ i k k ¯Z( u [ i ] , . . . , u k [ i k ]) , where u j [ i j ] means that the argument u j is repeated i j times. Moreover, the function ¯Z is epi-additive ineach variable. For i ∈ { , . . . , m } and u i +1 , . . . , u m ∈ Conv sc ( R n ) , the map u ¯Z( u [ i ] , u i +1 , . . . , u m ) is a continuous, epi-translation invariant valuation on Conv sc ( R n ) that is epi-homogeneous of degree i . The special case m = 1 in the previous result leads to the following result. Corollary 22. If Z :
Conv sc ( R n ) → R is a continuous and epi-translation invariant valuation that isepi-homogeneous of degree 1, then Z is epi-additive. Finally, we also obtain the dual statements. We say that a functional
Z :
Conv ( R n ; R ) → R is additive if Z( α v + β w ) = α Z( v ) + β Z( w ) for all α, β ≥ and v, w ∈ Conv ( R n ; R ) . Theorem 23.
Let
Z :
Conv ( R n ; R ) → R be a continuous, dually epi-translation invariant valuation thatis homogeneous of degree m ∈ { , . . . , n } . There exists a symmetric function ¯Z : ( Conv ( R n ; R )) m → R such that for k ∈ N , v , . . . , v k ∈ Conv ( R n ; R ) and λ , . . . , λ k ≥ , Z( λ v + · · · + λ k v k ) = X i ,...,i k ∈{ ,...,m } i + ··· + i k = m (cid:18) mi · · · i k (cid:19) λ i · · · λ i k k ¯Z( v [ i ] , . . . , v k [ i k ]) . Moreover, the function ¯Z is additive in each variable. For i ∈ { , . . . , m } and v i +1 , . . . , v m ∈ Conv ( R n ; R ) ,the map v ¯Z( v [ i ] , v i +1 , . . . , v m ) is a continuous and dually epi-translation invariant valuation on Conv ( R n ; R ) that is homogeneous of degree i . The special case m = 1 in the previous result leads to the following result. Corollary 24. If Z :
Conv ( R n ; R ) → R is a continuous and dually epi-translation invariant valuationthat is homogeneous of degree 1, then Z is additive. Let ζ ∈ C c ( R n ) . By Proposition 19, the functional Z( v ) = Z R n × R n ζ ( x ) dΘ ( v, ( x, y )) defines a continuous, dually epi-translation invariant valuation on Conv ( R n ; R ) that is homogeneous ofdegree n . Hence, by Theorem 23, for v , . . . , v k ∈ Conv ( R n ; R ) and λ , . . . , λ k ≥ , there exists asymmetric function ¯Z : ( Conv ( R n ; R )) n → R such that Z( λ v + · · · + λ k v k ) = X i ,...,i k ∈{ ,...,n } i + ··· + i k = n (cid:18) ni · · · i k (cid:19) λ i · · · λ i k k ¯Z( v [ i ] , . . . , v k [ i k ]) . If we assume in addition that v , . . . , v k ∈ C ( R n ) , then by (8) and properties of the mixed discriminant,we can also write Z( λ v + · · · + λ k v k ) = Z R n ζ ( x ) det(D ( λ v + · · · + λ k v k )( x )) d x = k X i ,...,i n =1 λ i · · · λ i n Z R n ζ ( x ) det(D v i ( x ) , . . . , D v i n ( x )) d x. It is now easy to see that for such functions v , . . . , v k and i , . . . , i k ∈ { , . . . , n } with i + · · · + i k = n , ¯Z( v [ i ] , . . . , v k [ i k ]) = Z R n ζ ( x ) det(D v ( x )[ i ] , . . . , D v k [ i k ]) d x. Note that this is a special case of (2).7. C
LASSIFICATION T HEOREMS
The classification of valuations that are epi-homogenous of degree 0 is straightforward.
Theorem 25.
A functional
Z :
Conv sc ( R n ) → R is a continuous and epi-translation invariant valuationthat is epi-homogeneous of degree , if and only if Z is constant.Proof. Let
Z :
Conv sc ( R n ) → R be a continuous and epi-translation invariant valuation that is epi-homogeneous of degree zero. We show that Z is constant. Indeed, for given y ∈ R n , the functional ˜Z y : K n → R defined by ˜Z y ( K ) = Z( ℓ y + I K ) . is a zero-homogeneous, continuous and translation invariant valuation on K n and therefore constant.Such a constant cannot depend on y , as, choosing K = { } , we obtain I { } + ℓ y = I { } + ℓ y for all y, y ∈ R n . Hence there exists α ∈ R such that Z( I K + ℓ y ) = α for all K ∈ K n and y ∈ R n . Thus the statement follows from applying Lemma 16 to Z − α . (cid:3) By duality, we also obtain the following result.
Theorem 26.
A functional
Z :
Conv ( R n ; R ) → R is a continuous and dually epi-translation invariantvaluation that is homogeneous of degree , if and only if Z is constant. Next, we prove Theorem 2. The “if” part of the proof follows from Proposition 20 and the subsequentremark. The proof of the theorem is completed by the next statement.
Proposition 27. If Z :
Conv sc ( R n ) → R is a continuous, epi-translation invariant valuation, that isepi-homogeneous of degree n , then there exists ζ ∈ C c ( R n ) such that Z( u ) = Z dom( u ) ζ ( ∇ u ( x )) d x for every u ∈ Conv sc ( R n ) .Proof. For y ∈ R n , we consider the map ˜Z y : K n → R defined by ˜Z y ( K ) = Z( ℓ y + I K ) . We know from the proof of Theorem 1 that ˜Z y is a continuous and translation invariant valuation on K n . Moreover, as the functional Z is epi-homogeneous of degree n , the functional ˜Z y is homogeneousof degree n . By Theorem 8, for each y ∈ R n , there exists a constant, that we denote by ζ ( y ) , such that(12) ˜Z( K ) = ζ ( y ) V n ( K ) for every K ∈ K n . As Z is continuous, the function ζ : R n → R is continuous. We prove, by contra-diction, that ζ has compact support. Assume that there exists a sequence y k ∈ R n , such that(13) lim k →∞ | y k | = + ∞ and ζ ( y k ) = 0 for every k . Without loss of generality, we may assume that(14) lim k →∞ y k | y k | = e n where e n is the n -th element of the canonical basis of R n .Let B k = { x ∈ y ⊥ k : | x | ≤ } , B ∞ = { x ∈ e ⊥ n : | x | ≤ } . Define the cylinder C k = (cid:26) x + ty k : x ∈ B k , t ∈ (cid:20) , ζ ( y k ) (cid:21)(cid:27) . We have V n ( C k ) = κ n − ζ ( y k ) , where κ n − is the ( n − -dimensional volume of the unit ball in R n − .For k ∈ N , we consider the function u k = ℓ y k + I C k . This is a sequence of functions in Conv sc ( R n ) ; using (13) and (14), it follows from Lemma 10 that u k epi-converges to u ∞ = I B ∞ . In particular, by the continuity of Z and (12) we get u ∞ ) = lim k →∞ Z( u k ) . On the other hand, by the definition of u k and (12), Z( u k ) = ζ ( y k ) V n ( C k ) = κ n − > . This completes the proof. (cid:3)
8. V
ALUATIONS WITHOUT V ERTICAL T RANSLATION I NVARIANCE
In this part we see that Theorems 1 and 4 are no longer true if we remove the assumption of verticaltranslation invariance. To do so, on the base of Theorem 17 we construct the following example. For η ∈ C c ( R n ) and v ∈ Conv ( R n ; R ) ∩ C ( R n ) , define(15) Z( v ) = Z R n e v ( x ) −h∇ v ( x ) ,x i η ( x ) det(D v ( x )) d x. By Theorem 17, the functional defined in (15) can be extended to a continuous valuation on Conv ( R n ; R ) .It is dually translation invariant but not vertically translation invariant. We choose v ∈ Conv ( R n ; R ) as v ( x ) = | x | . Note that the Hessian matrix of v is everywhere equal to the identity matrix. Hence det(D v ) = 1 on R n . For λ ≥ we have Z( λv ) = λ n Z R n η ( x ) e λ | x | d x. If η is non-negative and η ( x ) ≥ for every x such that ≤ | x | ≤ , then Z( λv ) ≥ cλ n e λ/ for a suitable constant c > and for every λ ≥ . Hence Z( λv ) does not have polynomial growth as λ tends to ∞ . Theorem 28.
There exist continuous, dually translation invariant valuations on
Conv ( R n ; R ) whichcannot be written as finite sums of homogeneous valuations. As a consequence we also have the following dual statement.
Theorem 29.
There exist continuous, translation invariant valuations on
Conv sc ( R n ) which cannot bewritten as finite sums of epi-homogeneous valuations.
9. E PI - TRANSLATION I NVARIANT V ALUATIONS ON C OERCIVE F UNCTIONS
In this part we prove that every continuous and epi-translation invariant valuation on Conv coe ( R n ) istrivial. Theorem 30.
Every continuous, epi-translation invariant valuation
Z :
Conv coe ( R n ) → R is constant.Proof. Let
Z :
Conv coe ( R n ) → R be a continuous, epi-translation invariant valuation. We need to showthat there exists α ∈ R such that Z( u ) = α for every u ∈ Conv coe ( R n ) . As in the proof of Theorem 1define for y ∈ R n \{ } the map ˜Z y : K n → R by ˜Z y ( K ) = Z( ℓ y + I K ) for every K ∈ K n . Since ˜Z y is a continuous and translation invariant valuation, by Theorem 7 it admitsa homogeneous decomposition ˜Z y = n X j =0 ˜Z y,j , where each ˜Z y,j is a continuous, translation invariant valuation on K n that is homogeneous of degree j .Next, we will show that ˜Z y,j ≡ for all ≤ j ≤ n . Since ˜Z y, ( K ) = lim λ → ˜Z y, ( λK ) = ˜Z y, ( { } ) for every K ∈ K n , this will then imply that ˜Z y is constant. By continuity it is enough to show that ˜Z y,j vanishes on polytopes for all ≤ j ≤ n . Since ˜Z y is continuous, it is enough to restrict to polytopes withno facet parallel to y ⊥ . Therefore, fix such a polytope P ∈ P n of dimension at least one. By translationinvariance we can assume that the origin is one of the vertices of P and that P lies in the half-space { x ∈ R n : h x, y i ≥ } . In particular, this gives P ∩ y ⊥ = { } , h x, y i > for all x ∈ P \{ } andmoreover h x, y i > and for all x ∈ λP \{ } for all λ > . Due to the choice of P we obtain that ℓ y + I λP is epi-convergent to ℓ y + I C as λ → ∞ where C is the infinite cone over P with apex at theorigin, that is C is the positive hull of P . Furthermore ℓ y + I C ∈ Conv coe ( R n ) since y = 0 . By continuitythis gives Z( ℓ y + I C ) = lim λ →∞ Z( ℓ y + I λP ) = lim λ →∞ ˜Z y ( λP ) = lim λ →∞ n X j =0 λ j ˜Z y,j ( P ) . Since the left side of this equation is finite, we have ˜Z y,n ( P ) = 0 . Otherwise, the right side wouldbe ±∞ , depending on the sign of ˜Z y,n ( P ) . Since P was arbitrary, we obtain that ˜Z y,n vanishes on allcompact convex polytopes of dimension greater or equal than and by continuity ˜Z y,n ≡ . Similarly,one can now show by induction that also ˜Z y,j ≡ for all ≤ j ≤ n − . We have proven so far that for every y ∈ R n \{ } there exists a constant α ( y ) ∈ R such that ˜Z y ≡ α ( y ) .Since α ( y ) = ˜Z y ( { } ) = Z( I { } ) , we obtain that α ( y ) is in fact independent of y , that is, there exists α ∈ R such that ˜Z y ≡ α for every y ∈ R n . By the definition of ˜Z y and the vertical translation invariance of ˜Z this gives Z( ℓ y + I K + β ) = α for every K ∈ K n , y ∈ R n \{ } and β ∈ R . The claim now follows from Lemma 16. (cid:3) If u ∈ Conv coe ( R n ) , then its conjugate u ∗ ∈ Conv ( R n ) and the origin is an interior point of its domain(see, for example, [35, Theorem 11.8]). Let Conv od ( R n ) be the set of functions in Conv ( R n ) with theorigin in the interior of its domain. Theorem 30 has the following dual. Theorem 31.
Every continuous, dually epi-translation invariant valuation
Z :
Conv od ( R n ) → R isconstant. R EFERENCES [1] S. Alesker,
Continuous rotation invariant valuations on convex sets , Ann. of Math. (2) (1999), 977–1005.[2] S. Alesker,
Description of translation invariant valuations on convex sets with solution of P. McMullen’s conjecture ,Geom. Funct. Anal. (2001), 244–272.[3] S. Alesker, Valuations on convex functions and convex sets and Monge–Amp`ere operators , Adv. Geom. (2019),313–322.[4] Y. Baryshnikov, R. Ghrist, and M. Wright, Hadwiger’s Theorem for definable functions , Adv. Math. (2013), 573–586.[5] G. Beer, R. T. Rockafellar and R. J.-B. Wets,
A characterization of epi-convergence in terms of convergence of levelsets , Proc. Amer. Math. Soc. (1992), 753–761.[6] A. Bernig and J. H. G. Fu,
Hermitian integral geometry , Ann. of Math. (2) (2011), 907–945.[7] A. Bernig, J. H. G. Fu, and G. Solanes,
Integral geometry of complex space forms , Geom. Funct. Anal. (2014),403–492.[8] L. Cavallina and A. Colesanti, Monotone valuations on the space of convex functions , Anal. Geom. Metr. Spaces (2015), 167–211.[9] A. Colesanti and I. Fragal`a, The first variation of the total mass of log-concave functions and related inequalities , Adv.Math. (2013), 708–749.[10] A. Colesanti and D. Hug,
Hessian measures of semi-convex functions and applications to support measures of convexbodies , Manuscripta Math. (2000), 209–238.[11] A. Colesanti and D. Hug,
Steiner type formulae and weighted measures of singularities for semi-convex functions , Trans.Amer. Math. Soc. (2000), 3239–3263.[12] A. Colesanti and N. Lombardi,
Valuations on the space of quasi-concave functions , Geometric aspects of functionalanalysis (B. Klartag and E. Milman, eds.), Lecture Notes in Math., Springer International Publishing, Cham, 2017,71–105.[13] A. Colesanti, N. Lombardi, and L. Parapatits,
Translation invariant valuations on quasi-concave functions , Studia Math. (2018), 79–99.[14] A. Colesanti, M. Ludwig, and F. Mussnig,
Minkowski valuations on convex functions , Calc. Var. Partial DifferentialEquations (2017), Art. 162, 29.[15] A. Colesanti, M. Ludwig, and F. Mussnig, Valuations on convex functions , Int. Math. Res. Not. IMRN (2019), 2384–2410.[16] A. Colesanti, M. Ludwig, and F. Mussnig,
Hessian valuations , Indiana Univ. Math. J., in press.[17] A. Colesanti, D. Pagnini, P. Tradacete, and I. Villanueva,
Dot product invariant valuations on Lip ( S n − ) ,arXiv:1906.04118 (2019).[18] H. Groemer, On the extension of additive functionals on classes of convex sets , Pacific J. Math. (1978), 397–410.[19] C. Haberl, Minkowski valuations intertwining with the special linear group , J. Eur. Math. Soc. (JEMS) (2012),1565–1597.[20] C. Haberl and L. Parapatits, The centro-affine Hadwiger theorem , J. Amer. Math. Soc. (2014), 685–705.[21] C. Haberl and L. Parapatits, Moments and valuations , Amer. J. Math. (2017), 1575–1603. [22] H. Hadwiger,
Vorlesungen ¨uber Inhalt, Oberfl¨ache und Isoperimetrie , Springer, Berlin, 1957.[23] D. A. Klain and G.-C. Rota,
Introduction to Geometric Probability , Cambridge University Press, Cambridge, 1997.[24] H. Kone,
Valuations on Orlicz spaces and L φ -star sets , Adv. in Appl. Math. (2014), 82–98.[25] J. Li and D. Ma, Laplace transforms and valuations , J. Funct. Anal. (2017), 738–758.[26] M. Ludwig,
Fisher information and valuations , Adv. Math. (2011), 2700–2711.[27] M. Ludwig,
Valuations on Sobolev spaces , Amer. J. Math. (2012), 827–842.[28] M. Ludwig and M. Reitzner,
A classification of
SL( n ) invariant valuations , Ann. of Math. (2) (2010), 1219–1267.[29] D. Ma, Real-valued valuations on Sobolev spaces , Sci. China Math. (2016), 921–934.[30] P. McMullen, Valuations and Euler-type relations on certain classes of convex polytopes , Proc. London Math. Soc. (3) (1977), 113–135.[31] V. D. Milman and L. Rotem, Mixed integrals and related inequalities , J. Funct. Anal. (2013), 570–604.[32] F. Mussnig,
Valuations on log-concave functions , arXiv:1707.06428 (2017).[33] F. Mussnig, SL ( n ) invariant valuations on super-coercive convex functions , arXiv:1903.04225 (2019).[34] F. Mussnig, Volume, polar volume and Euler characteristic for convex functions , Adv. Math. (2019), 340–373.[35] R. T. Rockafellar and R. J.-B. Wets,
Variational Analysis , Grundlehren der Mathematischen Wissenschaften, vol. 317,Springer-Verlag, Berlin, 1998.[36] R. Schneider,
Convex Bodies: the Brunn-Minkowski Theory , Second expanded ed., Encyclopedia of Mathematics andits Applications, vol. 151, Cambridge University Press, Cambridge, 2014.[37] P. Tradacete and I. Villanueva,
Valuations on Banach lattices , Int. Math. Res. Not., in press.[38] A. Tsang,
Valuations on L p spaces , Int. Math. Res. Not. (2010), 3993–4023.[39] T. Wang, Semi-valuations on
BV( R n ) , Indiana Univ. Math. J. (2014), 1447–1465.D IPARTIMENTO DI M ATEMATICA E I NFORMATICA “U. D
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