Integral Geometry and Algebraic Structures for Tensor Valuations

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1 Itegral Geometry ad Algebraic Structures for Tesor Valuatios Adreas Berig ad Daiel Hug Abstract I this survey, we cosider various itegral geometric formulas for tesorvalued valuatios that have bee obtaied by differet methods. Furthermore we explai i a iformal way recetly itroduced algebraic structures o the space of traslatio ivariat, smooth tesor valuatios, icludig covolutio, product, Poicaré duality ad AleserFourier trasform, ad their relatio to iematic formulas for tesor valuatios. I particular, we describe how the algebraic viewpoit leads to ew itersectioal iematic formulas ad substatially simplified Crofto formulas for traslatio ivariat tesor valuatios. We also highlight the coectio to geeral itegral geometric formulas for area measures. 1 Itroductio A importat part of itegral geometry is devoted to the ivestigatio of itegrals (mea values of the form ϕ(k gl µ(dg, G where K, L R are sets from a suitable itersectio stable class of sets, G is a group actig o R ad thus o its subsets, µ is a Haar measure o G, ad ϕ is a fuctioal with values i some vector space W. Commo choices for W are the reals or the space of siged Rado measures. Istead of the itersectio, Miowsi additio is aother atural choice for a set operatio which has bee studied. The priciple aim the is to express such itegrals by meas of basic geometric fuctioals of K Adreas Berig Istitut für Mathemati, GoetheUiversität Frafurt, RobertMayerStr. 10, Frafurt, Germay, Daiel Hug Karlsruhe Istitute of Techology (KIT, Departmet of Mathematics, D7618 Karlsruhe, Germay, 59
2 60 Adreas Berig ad Daiel Hug ad L. Depedig o the specific framewor, such as the class of sets or the type of fuctioal uder cosideratio, differet methods have bee developed to establish itegral geometric formulas, ragig from classical covexity, differetial geometry, geometric measure theory to the theory of valuatios. The iterplay betwee the theory of valuatios ad itegral geometry, although a classical topic i covexity, has bee expaded ad deepeed cosiderably i recet years. I the preset survey, we explore the itegral geometry of tesorvalued fuctioals. This study suggests ad requires geeralizatios i the theory of valuatios which are of idepedet iterest. Therefore, we describe how some algebraic operatios ow for smooth traslatio ivariat scalarvalued valuatios (product, covolutio, AleserFourier trasform ca be exteded to smooth traslatio ivariat tesorvalued valuatios. Although these extesios are straightforward to defie, they ecode various itegral geometric formulas for tesor valuatios, lie Croftotype formulas, rotatio sum formulas (also called additive iematic formulas ad itersectioal iematic formulas. Eve i the easiest case of traslatio ivariat ad O(covariat tesor valuatios, explicit formulas are hard to obtai by classical methods. With the preset algebraic approach, we are able to simplify the costats i Croftotype formulas for tesor valuatios, ad to formulate a ew type of itersectioal iematic formulas for tesor valuatios. For the latter we show how such formulas ca be explicitly calculated i the O(covariat case. As a importat byproduct, we compute the AleserFourier trasform o a certai class of smooth valuatios, called spherical valuatios. This result is of idepedet iterest ad is the techical heart of the computatio of the product of tesor valuatios. Tesor Valuatios The preset chapter is based o the geeral itroductio to valuatios i Chap. 1 ad o the descriptio ad structural aalysis of tesor valuatios cotaied i Chap.. The algebraic framewor for the ivestigatio of scalar valuatios, which has already proved to be very useful i itegral geometry, is outlied i Chap. 4. I these chapters relevat bacgroud iformatio is provided, icludig refereces to previous wor, motivatio ad hits to applicatios. The latter are also discussed i other parts of this volume, especially i Chaps. 10, 1 ad 13. Let us fix our otatio ad recall some basic structural facts. We will write V for a fiitedimesioal real vector space. Sometimes we fix a Euclidea structure o V, which allows us to idetify V with Euclidea space R. The space of compact covex sets (icludig the empty set is deoted by K (V (or K if V = R. The vector space of traslatio ivariat, cotiuous scalar valuatios is deoted by Val(V (or simply by Val if the vector space V is clear from the cotext. The smooth valuatios i Val(V costitute a importat subspace for which we write Val (V ; see Defiitio 4.5 ad Remar??, Defiitio 5.5 ad Propositio 5.8, ad Sectio 6.3. There is a atural decompositio of Val(V (ad the also of Val (V
3 Itegral Geometry ad Algebraic Structures for Tesor Valuatios 61 ito subspaces of differet parity ad differet degrees of homogeeity, hece if V has dimesio, ad similarly Val(V = Val ε m(v, m=0 ε=± Val (V = m=0 ε=± Val ε, m (V ; see Sects. 1.4, 4.1 ad Theorem 5.1. Our mai focus will be o valuatios with values i the space of symmetric tesors of a give ra p N 0, for which we write Sym p R or simply Sym p if the uderlyig vector space is clear from the cotext (resp., Sym p V i case of a geeral vector space V. Here we deviate from the otatio T p used i Chaps. 1 ad. The spaces of symmetric tesors of differet ras ca be combied to form a graded algebra i the usual way. By a tesor valuatio we mea a valuatio o K (V with values i the vector space of tesors of a fixed ra, say Sym p (V. For the space of traslatio ivariat, cotiuous tesor valuatios with values i Sym p (V we write TVal p (V ; cf. the otatio i Chapters 4, 6 ad Defiitio This vector space ca be idetified with Val(V Sym p (V (or Val Sym p, for short. If we restrict to smooth tesor valuatios, we add the superscript, that is TVal p, (V. It is clear that McMulle s decompositio exteds to tesor valuatios, hece TVal p (V = m=0 ε=± TVal p,ε m (V, if dim(v =. The correspodig decompositio is also available for smooth tesorvalued valuatios or valuatios covariat (or ivariat with respect to a compact subgroup G of the orthogoal group which acts trasitively o the uit sphere. The vector spaces of tesor valuatios satisfyig a additioal covariace coditio with respect to such a group G is fiitedimesioal ad cosists of smooth valuatios oly (cf. Example 4.6 ad Theorem I the followig, we will oly cosider rotatio covariat valuatios (see Chapter..1 Examples of Tesor Valuatios I the followig, we maily cosider traslatio ivariat tesor valuatios. However, we start with recallig geeral Miowsi tesors, which are traslatio covariat but ot ecessarily traslatio ivariat. For Miowsi tesors, ad hece for all isometry covariat cotiuous tesor valuatios, we first state a geeral Crofto
4 6 Adreas Berig ad Daiel Hug formula. The maor part of this cotributio is the devoted to traslatio ivariat, rotatio covariat, cotiuous tesor valuatios. I this framewor, we explai how algebraic structures ca be itroduced ad how they are related to Crofto formulas as well as to additive ad itersectioal iematic formulas. Crofto formulas for tesorvalued curvature measures are the subect of Chap.??. For {0,..., 1} ad K K, let Λ 0 (K,,..., Λ 1 (K, deote the support measures associated with K (see Sect They are Borel measures o Σ := R S 1 which are cocetrated o the ormal budle c K of K. Let κ deote the volume of the uit ball ad ω = κ the volume of its boudary, the uit sphere. Usig the support measures, we recall from Sects. 1.3 or.1 that the Miowsi tesors are defied by Φ r,s 1 (K := r!s! ω ω +s for {0,..., 1} ad r, s N 0, ad Φ r,0 (K := 1 x r dx. r! K Σ xr u s Λ (K, d(x, u, I additio, we defie Φ r,s := 0 for all other choices of idices. Clearly, the tesor valuatios Φ 0,s ad Φ 0,0, which are obtaied by choosig r = 0, are traslatio ivariat. However, these are ot the oly traslatio ivariat examples, sice e.g., for {1,..., }, also satisfies Φ1,1 1 (K + t = Φ1,1 1 (K for all K K ad t R. Further examples of cotiuous, isometry covariat tesor valuatios are obtaied by multiplyig the Miowsi tesors with powers of the metric tesor Q ad by taig liear combiatios. As show by Aleser [1, ], o other examples exist (see also Theorem.5. I the followig, we write Φ 1,1 1 Φ s (K : = Φ0,s (K = 1 ω s! ω us Λ (K, d(x, u +s Σ ( 1 1 = ω +s s! us S (K, du, S 1 for {0,..., 1}, where we used the th area measure S (K, of K, a Borel measure o S 1 defied by S (K, := κ ( Λ (K, R. I additio, we defie Φ 0 := V ad Φ s := 0 for s > 0. The ormalizatio is such that Φ 0 = V, for {0,..., }, where V is the th itrisic volume. Clearly, the tesor valuatios Q i Φ s, for {0,..., } ad i, s N 0, are cotiuous, traslatio ivariat, O(covariat, homogeeous of degree ad have tesor ra i + s. We have Φ s 0 for s 0, ad Φ0 s (K is idepedet of K. Hece, we usually
5 Itegral Geometry ad Algebraic Structures for Tesor Valuatios 63 exclude these trivial cases. Apart from these, Aleser showed that for each fixed {1,..., 1} the valuatios Q i Φ s, i, s N 0, i + s = p, s 1, form a basis of the vector space of all cotiuous, traslatio ivariat, O(covariat tesor valuatios of ra p which are homogeeous of degree. The fact that these valuatios spa the correspodig vector space is implied by [1, Prop. 4.9] (ad [], the proof is based i particular o basic represetatio theory. A result of Weil [17, Thm. 3.5] states that differeces of area measure of order, for ay fixed {1,..., d 1}, are dese i the vector space of differeces of fiite, cetered Borel measures o the uit sphere. From this the asserted liear idepedece of the tesor valuatios ca be iferred. We also refer to Sect. 6.5 where the preset case is discussed as a example of a very geeral represetatio theoretic theorem. The situatio for geeral tesor valuatios (which are ot ecessarily traslatio ivariat is more complicated. As explaied i Chap., the valuatios Q i Φ r,s spa the correspodig vector space, but there exist liear depedeces betwee these fuctioals. Although all liear relatios are ow ad the dimesio of the correspodig vector space (for fixed ra ad degree of homogeeity has bee determied, the situatio here is ot perfectly uderstood. I the followig, it will ofte (but ot always be sufficiet to eglect the metric tesor powers Q i ad ust cosider the tesor valuatios Φ s, sice the metric tesor commutes with the algebraic operatios to be cosidered.. Itegral Geometric Formulas Let A(,, for {0,..., }, deote the affie Grassmaia of flats i R, ad let µ deote the motio ivariat measure o A(, ormalized as i [13, 14]. The Crofto formulas to be discussed below relate the itegral mea Φ r,s (K E µ (de A(, of the tesor valuatio Φ r,s (K E of the itersectio of K with flats E A(, to tesor valuatios of K. Guessig from the scalar case, oe would expect that oly tesor valuatios of the form Q i Φ r,s + (K are required. It turs out, however, that for geeral r the situatio is more ivolved. The followig Crofto formulas for Miowsi tesors have bee established i [7]. Sice Φ r,s (K E = 0 if <, we oly have to cosider the cases where. We start with the basic case =, i which the Crofto formula has a particularly simple form. Theorem.1. For K K, r, s N 0 ad 0 1,
6 64 Adreas Berig ad Daiel Hug α Φ r,s,,s (K E µ Q s Φ r,0 (K, if s is eve, (de = A(, 0, if s is odd, where 1 Γ ( ( α,,s := (4π s ( s Γ +s! Γ ( ( +s Γ. This result essetially follows from Fubii s theorem, combied with a relatio due to McMulle, which coects the Miowsi tesors of K E ad the Miowsi tesors of K E, defied with respect to the flat E as the ambiet space (see (4 for a precise statemet. The mai case < is cosidered i the ext theorem. Theorem.. Let K K ad,, r, s N 0 with 0 < 1. The Φ r,s (K E µ (de A(, = s χ (1,,,s,z Qz Φ r,s z z=0 + (K + s 1 z=0 χ (,,,s,z Qz s z 1 ( πlφ r+s z l,l + s+z+l (K QΦr+s z l,l + s+z+l (K, (1 l=0 where χ (1,,,s,z ad χ(,,,s,z are explicitly ow costats. The costats χ (1,,,s,z ad χ(,,,s,z oly deped o the idicated lower idices. It is remarable that they are idepedet of r. Moreover, the righthad side of this Crofto formula also ivolves other tesor valuatios tha Φ r,s + (K. For istace, i the special case where = 3, =, = 0, r = 1 ad s =, Theorem. yields that Φ 1, 0 (K E µ (de = 1 3 Φ1, 1 (K + 1 4π QΦ1,0 1 (K Φ,1 0 (K. A(3, It ca be show that it is ot possible to write Φ,1 0 as a liear combiatio of Φ 1, 1 ad QΦ 1,0 1, which are the oly other Miowsi tesors of ra 3 ad homogeeity degree. The explicit expressios obtaied for the costats χ (1,,,s,z ad χ(,,,s,z i [7] require a multiple (fivefold summatio over products ad ratios of biomial coefficiets ad Gamma fuctios. Some progress which ca be made i simplifyig this represetatio is described i Chap.??. Sice the tesor valuatios o the righthad side of the Crofto formula (1 are ot liearly idepedet, the specific represetatio is ot uique. Usig the liear relatio due to McMulle, the result ca also be expressed i the form
7 Itegral Geometry ad Algebraic Structures for Tesor Valuatios 65 Φ r,s (K E µ (de A(, = s χ (1,,,s,z Qz Φ r,s z z=0 l s z + (K + s 1 z=0 χ (,,,s,z Qz ( QΦ r+s z l,l + s+z+l (K πlφr+s z l,l + s+z+l (K with the same costats as before. From ( we ow deduce the Crofto formula for the traslatio ivariat tesor valuatios Φ s. For r = 0, the sum l s z o the righthad side of ( is ozero oly if l = s z. Therefore, after some idex shift (ad discussio of the boudary cases z = 0 ad z = s, we obtai for <, where A(, s Φ s (K E µ (de = χ (,,,s,z Qz Φ+ s z (K (3 z=0 χ (,,,s,z = χ(1,,,s,z + χ(,,,s,z 1 π(s zχ(,,,s,z. Sice the righthad side of (3 is uiquely determied by the lefthad side ad the tesor valuatios o the righthad side are liearly idepedet, the costat χ (,,,s,z is uiquely determied. Usig the expressio which is obtaied for χ(,,,s,z from the costats χ (1,,,s,z ad χ(,,,s,z provided i [7], it seems to be a formidable tas to get a reasoably simple expressio for this costat. If =, the Theorem.1 shows that (3 remais true if we defie χ,,,s, s := α,,s if s is eve, ad as zero i all other cases. As we will see, the approach of algebraic itegral geometry to (3 will reveal that χ (,,,s,z has ideed a surprisigly simple expressio. To compare the algebraic approach with the oe used i [7], ad exteded to tesorial curvature measures i Chap.??, we poit out that the result of Theorem. is complemeted by ad i fact is based o a itrisic Crofto formula, where the tesor valuatio Φ r,s (K E is replaced by Φ r,s,e (K E. The latter is the tesor valuatio of the itersectio K E, determied with respect to E as the ambiet space but cosidered as a tesor i R (see Sectio?? or [7] for a explicit defiitio. The two tesors are coected by the relatio Φ r,s Q(E (K E = m m 0 (4π m m! Φr,s m,e (K E, (4 due to McMulle [11, Theorem 5.1] (see also [7], where Q(E is the metric tesor of the liear subspace orthogoal to the directio space of E but agai cosidered as a tesor i R, that is, Q(E = e e, where e +1,..., e is a orthoormal basis of E. Note that for s = 0 we get Φ r,0 (K E = Φ r,0,e (K E, sice the (
8 66 Adreas Berig ad Daiel Hug itrisic volumes ad the suitably ormalized curvature measures are idepedet of the ambiet space. The itrisic Crofto formula for Φ r,s,e (K E µ (de A(, has the same structure as the extrisic Crofto formula stated i Theorem., but the costats are differet. Apart from reducig the umber of summatios required for determiig the costats, progress i uderstadig the structure of these (itrisic ad extrisic itegral geometric formulas ca be made by localizig the Miowsi tesors. This is the topic of Chapter??. Crofto ad itersectioal iematic formulas for Miowsi tesors Φ r,s with s = 0 are special cases of correspodig (more geeral itegral geometric formulas for curvature measures. For example, we have ad A(, Φ r,0 (K E µ (de = a Φ r,0 + (K (5 Φ r,0 (K gm µ(dg = G a Φ r,0 + (KV (M, (6 = where G is the Euclidea motio group, µ is the suitably ormalized Haar measure ad the (simple costats a are ow explicitly. Therefore, we focus o the case s 0 (ad r = 0 i the followig. A close coectio betwee Crofto formulas ad itersectioal iematic formulas follows from Hadwiger s geeral itegral geometric theorem (see [14, Theorem 5.1.]. It states that for ay cotiuous valuatio ϕ o the space of covex bodies ad for all K, M K, we have G ϕ(k gm µ(dg = ϕ(k E µ (dev (M. (7 =0 A(, Hece, if a Crofto formula for the fuctioal ϕ is available, the a itersectioal iematic formula is a immediate cosequece. This statemet icludes also tesorvalued fuctioals, sice (7 ca be applied coordiatewise. I particular, this shows that (6 ca be obtaied from (5. I the same way, Theorem. ad the special case show i (3 imply iematic formulas for itersectios of covex bodies, oe fixed the other movig. Thus, for istace, we obtai G Φ s (K gm µ(dg = s z=0 = χ (,,,s,z Qz Φ s z + (KV (M. (8 These results are related to ad i fact ispired geeral itegral geometric formulas for area measures (see [10]. The startig poit is a local versio of Hadwiger s geeral itegral geometric theorem for measurevalued valuatios. To state it, let
9 Itegral Geometry ad Algebraic Structures for Tesor Valuatios 67 M + (S 1 be the coe of oegative measures i the vector space M (S 1 of fiite Borel measures o the uit sphere. Theorem.3. Let ϕ : K M + (S 1 be a cotiuous ad additive mappig with ϕ(/0, = 0 (the zero measure. The, for K, M K ad Borel sets A S 1, G ϕ(k gm, A µ(dg = =0 [T, ϕ(k, ](AV (M, (9 with (the Crofto operator T, : M + (S 1 M + (S 1 give by T, ϕ(k, := A(, ϕ(k E, µ (de, = 0,...,. We wat to apply this result to area measures of covex bodies, hece we eed a Crofto formula for area measures. The statemet of such a Crofto formula is based o Fourier operators I p, for p { 1, 0, 1,..., }, which act o C fuctios o S 1. For f C (S 1, let f p be the extesio of f to R \ {0} which is homogeeous of degree + p, ad let ˆf p be the distributioal Fourier trasform of f p. For 0 < p <, the restrictio I p ( f of ˆf p to the uit sphere is agai a smooth fuctio. Let Hs deote the space of spherical harmoics of degree s. Recall that a spherical harmoic of degree s is the restrictio to the uit sphere of a homogeeous polyomial p of degree s o R which satisfies p = 0 (ad hece is called harmoic, where is the Laplace operator. We refer to [13] for more iformatio o spherical harmoics. Sice I p itertwies the group actio of SO(, we have I p ( f s = λ s (, p f s for f s Hs ad some λ s (, p C. It is ow that Γ ( s+p λ s (, p = π p i s Γ ( s+ p Note that λ s (, p is purely imagiary if s is odd, ad real if s is eve. See [10] for a summary of the mai properties of this Fourier operator ad [9, 8] for a detailed expositio. Usig the coectio to mea sectio bodies (see [8] ad the Fourier operators I p, the followig Crofto formula for area measures has bee established i [10, Theorem 3.1]. Theorem.4. Let 1 < ad K K. The S (K E, µ (de = a(,, I I S + ( K, (10 with A(, a(,, := π (+/ ( +. Γ ( +1 Γ ( Γ ( +1 Γ (. Let I be the reflectio operator (I f (u = f ( u, u S 1, for a fuctio f o the uit sphere. The operator T,, := a(,, I I I, for 1 <, ad
10 68 Adreas Berig ad Daiel Hug the idetity operator T,, act as liear operators o M (S 1 ad ca be used to express (10 i the form S (K E, µ (de = T,, S + (K,. (11 A(, This is also true for = < if we defie ( 1 1 ω T,, S (K, := V (Kσ, where σ is spherical Lebesgue measure. Combiig equatios (9 ad (11, we obtai a iematic formula for area measures. Usig agai the operator T,,, it ca be stated i the form G S (K gm, A µ(dg = = ω [T,, S + (K, ](AV (M, for = 1,..., 1. Sice the Fourier operators act as multiplier operators o spherical harmoics, it follows that Theorem.4 ca be rewritte i the form A(, s(u S (K E, du µ (de S 1 = a s (,, f s(u S + (K, du, S 1 (1 where f s H s ad a s (s,, := a(,, b s (,, with ( ( Γ s+ b s (,, := π Γ s+. Γ Γ ( s+ ( s+ + I additio to Crofto ad itersectioal iematic formulas, there is aother classical type of itegral geometric formula. Sice they ivolve rotatios ad Miowsi sums of covex bodies, it is ustified to call them rotatio sum formulas. Let SO( deote the group of rotatios ad let ν deote the Haar probability measure o this group. A geeral form of such a formula ca agai be stated for area measures. Let K, M K be covex bodies ad let α, β S 1 be Borel sets. The [13, Theorem 4.4.6] ca be writte i the form 1 α(u1 β (ρ 1 u S (K + ρm, du ν(dρ SO( S 1 = 1 ( ω S (K, αs (M, β. (13 =0 More geerally, by the iversio ivariace of the Haar measure ν, by basic measure theoretic extesio argumets, ad by a applicatio of (13 to the coordiate
11 Itegral Geometry ad Algebraic Structures for Tesor Valuatios 69 fuctios of a arbitrary cotiuous fuctio f : S 1 S 1 Sym s 1 Sym s, for give s 1, s N 0, we obtai f (u, ρu S (K + ρ 1 M, du ν(dρ SO( S 1 = 1 ( ω =0 f (u, v ( S (K, S (M, (d(u, v. (S 1 To simplify costats, we defie Choosig f (u, v = u s 1 v s, we thus get SO( ad hece SO( φ s (K := S 1 us S (K, du. (14 (id s 1 ρ s φ s 1+s (K + ρ 1 M ν(dρ = SO( = 1 ω =0 us 1 (ρu s S (K + ρ 1 M, du ν(dρ S 1 ( (S 1 us 1 v s ( S (K, S (M, (d(u, v, (15 (id s 1 ρ s φ s 1+s (K + ρ 1 M ν(dρ = 1 ω =0 ( φ s 1 (K φ s (M. Up to the differet ormalizatio, this is the additive iematic formula for tesor valuatios stated i [6, Theorem 5]. I particular, SO( φ s (K + ρm ν(dρ = 1 ω =0 ( φ s (KS (M, where S i (M := S i (M, S 1 ( = κ 1Vi i i (M. I the followig sectio, we develop basic algebraic structures for tesor valuatios ad provide applicatios to itegral geometry. From this approach, we will obtai a Crofto formula for the tesor valuatios Φ s, but also for aother set of tesor valuatios, deoted by Ψ s, for which the Crofto formula has diagoal form. Moreover, we will study more geeral itersectioal iematic formulas tha the oe cosidered i (8 ad describe the coectio betwee itersectioal ad additive iematic formulas. I the course of our aalysis, we determie Aleser s Fourier operator for spherical valuatios, that is, valuatios obtaied by itegratio of a spherical harmoic (or, more geerally, ay smooth spherical fuctio agaist a area measure.
12 70 Adreas Berig ad Daiel Hug 3 Algebraic Structures o Tesor Valuatios Recall that Val = Val(R deotes the Baach space of traslatio ivariat cotiuous valuatios o V = R, ad Val = Val (R is the dese subspace of smooth valuatios; see Chapters 4 ad 5 for more iformatio. I this sectio, we first discuss the extesio of basic operatios ad trasformatios from scalar valuatios to tesorvalued valuatios. The scalar case is described i Chap. 4. I the followig, we usually wor i Euclidea space R with the Lebesgue measure ad the volume fuctioal V o covex bodies. Sice some of the results are also stated i ivariat terms, we write vol for a volume measure o V, that is, a choice of a traslatio ivariat locally fiite Haar measure o a dimesioal vector space V. Of course, i case V = R we always use V as a specific choice of the restrictio of a volume measure vol to K (the correspodig choice is made for V = R R. 3.1 Product Existece ad uiqueess of the product of smooth valuatios is provided by the followig result; see also Sect. 4. for the more geeral costructio of a exterior product betwee smooth scalarvalued valuatios o possibly differet vector spaces. Propositio.5. Let φ 1, φ Val be smooth valuatios o R give by φ i (K = vol(k + A i, K K, where A 1, A K are smooth covex bodies with positive Gauss curvature at every boudary poit. Let : R R R be the diagoal embeddig. The φ 1 φ (K := vol( K + A 1 A, K K, exteds by cotiuity ad biliearity to a product o Val. The product is compatible with the degree of a valuatio (i.e., if φ i has degree i, the φ 1 φ has degree 1 + if 1 +, ad more geerally with the actio of the group GL(. We ca exted the product compoetwise from smooth scalarvalued valuatios to smooth tesorvalued valuatios. To see this, let V be a fiitedimesioal vector space, V = R say, ad s 1, s N 0. Let Φ i TVal s i, (V for i = 1,. Let w 1,..., w m be a basis of Sym s 1V, ad let u 1,..., u l be a basis of Sym s V. The there are φ i, ψ Val (V, i {1,..., m} ad {1,..., l}, such that Φ 1 (K = m i=1 φ i (Kw i ad Φ (K = l ψ (Ku =1
13 Itegral Geometry ad Algebraic Structures for Tesor Valuatios 71 for K K (V. Now we would lie to defie (omittig the obvious rages of the idices (Φ 1 Φ (K := (φ i ψ (Kw i u. i, The dot o the righthad side is the product of the smooth valuatios φ i, ψ, ad w i u Sym s 1+s V deotes the symmetric tesor product of the symmetric tesors w i Sym s 1V ad u Sym s V. Let us verify that this defiitio is idepedet of the chose bases. For this, let w i = c i w with some ivertible matrix (c i, ad let u i = e i u with a ivertible matrix (e i. If Φ 1 (K = φ i (Kw i = φ i (Kw i, i i the a compariso of coefficiets yields that φ i = c i φ, where (c i deotes the matrix iverse. Similarly, from Φ (K = i ψ (Ku i = ψ i (Ku i, i we coclude that ψ i = e i ψ, where (e i deotes the matrix iverse. Therefore, we have ( (φ i ψ w iu = c a1i φ a1 e b 1 ψ b1 c ia w a e b u b i, i,,b 1,b a 1,b 1 a,b ( = c a1i c ia e b 1 e b (φ a1 ψ b1 w a u b a 1,a,b 1,b i, = (φ a ψ a w a u b, a,b }{{} =δ a 1 a δ b 1 b which proves the asserted idepedece of the represetatio. Thus, recallig that TVal s m(v deotes the vector space of traslatio ivariat cotiuous valuatios o K (V which are homogeeous of degree m ad tae values i the vector space Sym s V of symmetric tesors of ra s over V, ad that TValm s, (V is the subspace cosistig of the smooth elemets of this vector space, we have Φ 1 Φ TVal s 1+s, +l (V, + l, for Φ 1 TVal s 1, (V, Φ TVal s, l (V ad, l {0,..., }. A similar descriptio ad similar argumets ca be give for the operatios cosidered i the followig subsectios.
14 7 Adreas Berig ad Daiel Hug 3. Covolutio Similarly as for the product of valuatios, a explicit defiitio of the covolutio of two valuatios is give oly for a suitable subclass of valuatios (cf. Sect Propositio.6. Let φ 1, φ Val be smooth valuatios o R give by φ i (K = vol(k + A i, where A 1, A are smooth covex bodies with positive Gauss curvature at every boudary poit. The φ 1 φ (K := vol(k + A 1 + A, exteds by cotiuity ad biliearity to a product (which is called covolutio o Val. Writte i ivariat terms, the covolutio is a biliear map Val (V Des(V Val (V Des(V Val (V Des(V, where Des(V is the oedimesioal space of traslatio ivariat, locally fiite complexvalued Haar measures (Lebesgue measures, see Sect. 4.3 o the dual space V. It is compatible with the actio of the group GL( ad with the codegree of a valuatio (i.e., if φ i has degree i, the φ 1 φ has degree 1 + if 1 +. The covolutio ca be exteded compoetwise to a covolutio o the space of traslatio ivariat smooth tesor valuatios. Hece we have Φ 1 Φ TVal s 1+s, +l (V, + l, for Φ 1 TVal s 1, (V, Φ TVal s, l (V ad, l {0,..., }. This is aalogous to the defiitio ad computatio i the previous subsectio. 3.3 AleserFourier Trasform Aleser itroduced a operatio o smooth valuatios, ow called AleserFourier trasform (cf. Sect It is a map F : Val (R Val (R which reverses the degree of homogeeity, that is, F : Val (R Val (R, ad which trasforms product ito covolutio of smooth valuatios, more precisely, we have F(φ 1 φ = F(φ 1 F(φ. (16
15 Itegral Geometry ad Algebraic Structures for Tesor Valuatios 73 O valuatios which are smooth ad eve, the AleserFourier trasform ca easily be described i terms of Klai fuctios as follows. Let φ Val,+ (R (the space of smooth ad eve valuatios which are homogeeous of degree. The the restrictio of φ to a dimesioal subspace E is a multiple Kl φ (E of the volume, ad the resultig fuctio (Klai fuctio Kl φ determies φ. The Kl Fφ (E = Kl φ (E for every ( dimesioal subspace E. As a example (ad cosequece of the relatio to Klai fuctios, the itrisic volumes satisfy F(V = V, (17 where V 0,...,V deote the itrisic volumes o K. The descriptio i the odd case is more ivolved ad it is preferable to describe it i ivariat terms (i.e., without referrig to a Euclidea structure. Let V be a dimesioal real vector space. The F : Val (V Val (V Des(V, where Des deotes the oedimesioal space of desities (Lebesgue measures. This map commutes with the actio of GL(V o both sides. Applyig it twice (ad usig the idetificatio Des(V Des(V = C, it satisfies the Placherel type formula (F φ(k = φ( K. Worig agai o Euclidea space V = R, we ca exted the AleserFourier trasform compoetwise to a map F : TVal s, TVal s, such that F : TVal s, TVal s,. It is ot a easy tas to determie the Fourier trasform of valuatios other tha the itrisic volumes. 3.4 Example: Itrisic Volumes As a example, let us compute the Aleser product of itrisic volumes V 0,...,V i R. We complemet the defiitio of the itrisic volumes by V l := 0 for l < 0. Let vol = V deote the volume measure o K. Recall Steier s formula (1.16 which states that vol(k + rb = i=0 V i (Kκ i r i, r 0.
16 74 Adreas Berig ad Daiel Hug Now we fix r 0 ad s 0 ad defie the smooth valuatios φ 1 (K := vol(k + rb ad φ (K := vol(k + sb. The φ 1 φ (K = vol(k + rb + sb = vol(k + (r + sb = hece φ 1 φ = =0 ( i + V i κ i+ r i s. i, =0 i V (Kκ (r + s, O the other had, sice φ 1 = i=0 V iκ i r i ad φ = i=0 V iκ i s i, we obtai φ 1 φ = V i V κ i κ r i s. i, =0 Now we compare the coefficiet of r i s i these equatios ad get ( i + V i κ i+ = V i V κ i κ. i Writig i istead of i ad istead of, we obtai [ ] i V i V = V i i+, (18 where we used the flag coefficiet [ ] ( κ :=, κ κ {0,..., }. Taig AleserFourier trasform o both sides yields [ ] i + V i V = V i i+. (19 The computatio of covolutio ad product of tesor valuatios follows the same scheme: first oe computes the covolutio of tesor valuatios, which ca be cosidered easier. The oe applies the AleserFourier trasform to obtai the product. However, i the tesorvalued case it is much harder to write dow the AleserFourier trasform i a explicit way. This step is the techical heart of our approach.
17 Itegral Geometry ad Algebraic Structures for Tesor Valuatios Poicaré Duality The product of smooth traslatio ivariat valuatios as well as the covolutio both satisfy a versio of Poicaré duality, which moreover are idetical up to a sig. To state this more precisely, recall that the vector spaces Val = Val (R, {0, }, are oedimesioal ad spaed by the Eulercharacteristic χ = V 0 ad the volume fuctioal V = vol, that is, Val 0 = R χ ad Val = R vol. We deote by φ 0, φ R the compoet of φ Val of degree 0 ad, respectively. Propositio.7. The pairigs ad are perfect, that is, the iduced maps Val Val R, (φ 1, φ (φ 1 φ, Val Val R, (φ 1, φ (φ 1 φ 0, pd m, pd c : Val Val, are iective with dese image. Moreover, { pd pd c = m o Val +. pd m o Val. To illustrate this propositio ad to highlight the differece betwee the two pairigs, let us compute them o a easy example. Let φ i (K := vol(k + A i, where A i, i {1, }, are smooth covex bodies with positive Gauss curvature. The φ 1 φ (K = vol(k + A 1 + A, ad hece (φ 1 φ 0 = vol(a 1 + A. O the other had, φ 1 φ (K = vol ( K + A 1 A. Usig Fubii s theorem, oe rewrites this as φ 1 φ (K = φ ((x A 1 K dx. R Taig for K a large ball reveals that (φ 1 φ = φ ( A 1 = vol(a A 1. If A 1 = A 1, the φ 1 is eve ad both pairigs agree ideed. The extesio of Poicaré duality to tesorvalued valuatios is postpoed to Sect. 4.1 where it is required for the descriptio of the relatio betwee additive ad itersectioal iematic formulas for tesor valuatios. 3.6 Explicit Computatios i the O(Equivariat Case I this subsectio, we outlie the explicit computatio of product, covolutio ad AleserFourier trasform i the O(equivariat case. Depedig o the situatio,
18 76 Adreas Berig ad Daiel Hug we will either use the basis cosistig of the tesor valuatios Q i Φ s i or the basis cosistig of the tesor valuatios Q i Ψ s i. The latter are defied i the followig propositio. Propositio.8. The followig statemets hold. (i For 0 < ad s 1, defie s := Φs + ( 1 Γ ( +s Γ ( + s 1 =1 (4π!Γ ( +s Γ ( + s Φ s 1Q Ψ s ad let Ψ 0 := Φ 0. The Ψ s is the trace free part of Φs. I particular, Ψ s Φs mod Q. (ii For 0 < ad s 1, Φ s s ca be writte i terms of Ψ as Φ s = Ψ s + s =1 Γ ( +s Γ ( + s (4π!Γ ( +s Γ ( + s Ψ s Q. The iversio which is eeded to derive (ii from (i ca be accomplished with the help of Zeilberger s algorithm. The first ad easier step i the explicit calculatios of algebraic structures for tesor valuatios is to compute the covolutio of two tesor valuatios. Sice Φ s is smooth (i.e., each compoet is a smooth valuatio, we may write Φ s (K = ω,s, c(k where ω,s is a smooth ( 1form o the sphere budle R S 1 with values i Sym s R. Next, for valuatios represeted by differetial forms, there is a easy formula for the covolutio, which ivolves oly some liear ad biliear operatios (a id of Hodge star ad a wedge product. The resultig formula states that, for, l with + l ad s 1, s 1, we have ( ( l ω s1 + ω s + l l ( ( l s1 + s Φ s 1 Φ s l = ω s 1 +s + l s 1 (s1 1(s 1 1 s 1 s Φ s 1+s +l, or, usig the ormalizatio (14 which is more coveiet for this purpose, φ s 1 φ s l = ( +l ( +l (s 1 1(s 1 φ s 1+s 1 s 1 s +l. The computatio of the AleserFourier trasform of tesor valuatios is the mai step ad will be explaied i the ext subsectio. For 0 ad s 1, the result is
19 Itegral Geometry ad Algebraic Structures for Tesor Valuatios 77 F(Ψ s = is Ψ s, s F(Φ s = ( 1 is =0 (4π! Q Φ s. Fially, the product of two tesor valuatios ca be computed oce the covolutio ad the AleserFourier trasform are ow, see (16. The result is a bit more ivolved tha the formulas for covolutio ad AleserFourier trasform. The reaso is that the formula for the covolutio is best described i terms of the tesor valuatios Φ s, while the descriptio of the AleserFourier trasform has a simpler expressio for the Ψ s. After some algebraic maipulatios (which mae use of Zeilberger s algorithm, we arrive at Φ s 1 Φ s l = l + l ( + l ( 1 a m ( a m s 1 +s a=0 ( m 1 a mi{m, (4π a!( s 1 } a m=0 ( s1 + s m s 1 i a s 1 +s 1 ωs1 +s m++l i ω s1 i+ω s m+i+l (s 1 i 1(s m + i 1 1 s 1 s + m i=max{0,m s } Q a Φ s 1+s a +l. (0 Here 0, l with + l ad s 1, s 1. It seems that there is o simple closed expressio for the ier sum. 3.7 Tesor Valuatios Versus ScalarValued Valuatios The iterplay betwee tesor valuatios ad scalarvalued valuatios will be essetial i the computatio of the AleserFourier trasform. We therefore explai this ow is some more detail. We first eed some facts from represetatio theory. It is wellow that equivalece classes of complex irreducible (fiitedimesioal represetatios of SO( are idexed by their highest weights. The possible highest weights are tuples (λ 1, λ,..., λ of itegers such that 1. λ 1 λ... λ 0 if is odd,. λ 1 λ... λ 0 if is eve. Give λ = (λ 1,..., λ satisfyig this coditio, we will deote the correspodig equivalece class of represetatios by Γ λ. The decompositio of the SO(module Val has recetly bee obtaied i [3]. Theorem.9 ([3]. There is a isomorphism of SO(modules
20 78 Adreas Berig ad Daiel Hug Val = Γ λ, where λ rages over all highest weights such that λ, λ i 1 for all i ad λ i = 0 for i > mi{, }. I particular, these decompositios are multiplicityfree. Let Γ be a irreducible represetatio of SO( ad Γ its dual. The space of  homogeeous SO(equivariat Γ valued valuatios (i.e., maps Φ : K Γ such that Φ(gK = gφ(k for all g SO( is (Val Γ SO( = Hom SO( (Γ, Val. By Theorem.9, Γ appears i the decompositio of Val precisely if Γ appears, ad i this case the multiplicity is 1. By Schur s lemma it follows that dim(val Γ SO( = 1 i this case. Let us costruct the (uique up to scale equivariat Γ valued valuatio explicitly. Deote by Val (Γ the Γ isotypical summad, which is isomorphic to Γ sice Val is multiplicity free. Let φ 1,..., φ m be a basis of Val (Γ. These elemets play two differet roles: first we ca loo at them as valuatios, i.e., elemets of Val. Secod, we may thi of φ 1,..., φ m as basis of the irreducible represetatio Γ. The actio of SO( o this basis is give by gφ i = c i (gφ, where (c i (g i, is a matrix depedig o g. The map g (c i (g i, is a homomorphism of Lie groups SO( GL(m. Let φ1,..., φ m be the dual basis of Γ. The gφ i = (c i (g t φ = λ c i (g 1 φ, Usig the double role played by the φ i metioed above, we set Φ(K := φ i (Kφi Γ. (1 i We claim that Φ is a O(equivariat valuatio with values i Γ. Ideed, we compute
21 Itegral Geometry ad Algebraic Structures for Tesor Valuatios 79 Φ(gK = φ i (gkφi i = (g 1 φ i (Kφi i = c i (g 1 φ (Kφi i, = φ (K c i (g 1 φi i = φ (Kgφ = g(φ(k. Coversely, we ow start with a equivariat Γ valued cotiuous traslatio ivariat valuatio Φ of degree. Let w 1,..., w m be a basis of Γ. The we may loo at the compoets of Φ, i.e., we decompose Φ(K = φ i (Kw i i with φ i Val. Let the actio of SO( o Γ be give by We have Φ(gK = i g(φ(k = gw i = a i (gw. φ i (gkw i = (g 1 φ i (Kw i, i φ (Kgw = φ (Ka i (gw i. i, Comparig coefficiets yields g 1 φ i = a i (gφ, or gφ i = a i (g 1 φ. This shows that the subspace of Val spaed by φ 1,..., φ m is isomorphic to Γ. I summary, we have show the followig fact. Each SO(irreducible represetatio Γ appearig i the decompositio of Val correspods to the (uique up to scale Γ valued cotiuous traslatio ivariat valuatio Φ from (1. Coversely, the coefficiets of a Γ valued cotiuous traslatio ivariat valuatio spa a subspace of Val isomorphic to Γ. Let us ow discuss the special case of symmetric tesor valuatios. The SO( represetatio space Sym s is (i geeral ot irreducible. Ideed, the trace map tr : Sym s Sym s commutes with SO(, hece its erel is a ivariat subspace.
22 80 Adreas Berig ad Daiel Hug This subspace turs out to be the irreducible represetatio Γ (s,0,...,0 ad ca be idetified with the space Hs of spherical harmoics of degree s. Sice the trace map is oto, we get the followig decompositio. Sym s = Hs. Istead of studyig Sym s valued valuatios, we ca therefore study Hs valued valuatios. For s 1 ad 1 1, the represetatio Hs appears i Val with multiplicity 1. Sice Hs is selfdual, the costructio setched above yields i the special case Γ := Hs a uique (up to scale Hs valued equivariat cotiuous traslatio ivariat valuatio homogeeous of degree, which we deoted by Ψ s. 3.8 The AleserFourier Trasform As we have see i the previous subsectio, the study of (symmetric tesor valuatios ad the study of the H s isotypical summad of Val are equivalet. For the computatio of the AleserFourier trasform, it is easier to wor with scalarvalued valuatios. Let us first defie a particular class of valuatios, called spherical valuatios. Let f be a smooth fuctio o S 1. For {0,..., 1}, we defie a valuatio µ, f Val (R by ( 1 1 µ, f (K := ω S 1 f (y S (K, dy. Such valuatios are called spherical (see also the recet preprit [15]. Here the ormalizatio is chose such that for f 1 we have µ, f = V, {0,..., 1}. By Subsectio 3.7, the compoets of a SO(equivariat tesor valuatio are spherical. Sice the AleserFourier trasform of such a tesor valuatio is defied compoetwise, it suffices to compute the AleserFourier trasform of spherical valuatios. I this subsectio, we setch this (rather ivolved computatio. The first ad easy observatio is that, by Schur s lemma, there exist costats c,,s C which oly deped o,, s such that F(µ, f = c,,s µ, f, f H s. ( The multipliers c,,s of the AleserFourier trasform ca be computed i the eve case (i.e., if s is eve by looig at Klai fuctios. I the odd case, there seems to be o easy way to compute them. We adapt ideas from [1], where the multipliers of the αcosie trasform were computed, to our situatio. The mai poit is that the AleserFourier trasform is ot oly a SO(equivariat operator, but (if writte i itrisic terms is equivariat uder the larger group GL(. Usig elemets from
23 Itegral Geometry ad Algebraic Structures for Tesor Valuatios 81 the Lie algebra gl( allows us to pass from oe irreducible SO(represetatio to aother ad to obtai a recursive formula for the costats c,,s, which states that c,,s+ c,,s = + s + s. (3 This step requires extesive computatios usig differetial forms, ad we refer to [6] for the details. Next, oe ca use iductio over s,, to prove that c,,s = i s Γ ( ( Γ s+. Γ ( Γ ( s+ More precisely, i the eve case, we may use as iductio start the case s = 0, which correspods to itrisic volumes, whose AleserFourier trasform is ow by (17. I the odd case, we use as iductio start s = 3. I order to compute c,,3, we use a special case of a Crofto formula from [7] (see also Chap. to compute the quotiets c,+1,3 c. This fixes all costats up to a scalig which may deped o.,,3 More precisely, c,,s = ε i s Γ ( ( Γ s+ Γ ( Γ ( s+, (4 where ε depeds oly o. Usig fuctorial properties of the AleserFourier trasform, we fid that ε is idepedet of. I the twodimesioal case, however, there is a very explicit descriptio of the AleserFourier trasform (see also Example 4.14 (4 which fially allows us to deduce that ε = 1 for all. A alterative approach to determiig the costats c,,s is to prove idepedetly a Crofto formula for the tesor valuatios Ψ s, via the Crofto formula for area measures, as described before (see also Remar 4.6 i [10]. This poit of view suggests to relate the Fourier operator for spherical valuatios to the Fourier operators for spherical fuctios via the relatio F( µ, f = (π d µ d,i f, for f C (S d 1, where ( d 1 µ, f (K = (π f (u S (K, du, S d 1 is ust a reormalizatio of µ, f (K.
24 8 Adreas Berig ad Daiel Hug 4 Kiematic Formulas I this sectio, we first describe the iterplay betwee algebraic structures ad iematic formulas i geeral (i.e., for tesor valuatios which are equivariat uder a group G actig trasitively o the uit sphere. The we will specialize to the O(covariat case. 4.1 Relatio Betwee Kiematic Formulas ad Algebraic Structures Let G be a subgroup of O( which acts trasitively o the uit sphere. The the space TVal s,g (V of Gcovariat, traslatio ivariat cotiuous Sym s (V valued valuatios is fiitedimesioal. Next we defie two itegral geometric operators. We start with the oe for rotatio sum formulas. Let Φ TVal s 1+s,G (V. We defie a bivaluatio with values i the tesor product Sym s 1 V Sym s V by a G s 1,s (Φ(K, L := (id s 1 g s Φ(K + g 1 L ν(dg G for K, L K (V, where G is edowed with the Haar probability measure ν (see [16]. (This otatio is cosistet with the case V = R ad G = O(. Let Φ 1,..., Φ m1 be a basis of TVal s1,g (V, ad let Ψ 1,...,Ψ m be a basis of TVal s,g (V. Arguig as i the classical Hadwiger argumet (cf. [16, Theorem 4.3], it ca be see that there are costats c Φ i such that a G s 1,s (Φ(K, L = c Φ i Φ i (K Ψ (L i, for K, L K (V. The additive iematic operator is the map a G s 1,s : TVal s 1+s,G (V TVal s1,g (V TVal s,g (V Φ c Φ i Φ i Ψ, i, which is idepedet of the choice of the bases. I view of itersectioal iematic formulas, we defie a bivaluatio with values i Sym s 1 V Sym s V by s G 1,s (Φ(K, L := (id s 1 g s Φ(K ḡ 1 L µ(dḡ Ḡ for K, L K (V, where Ḡ is the group geerated by G ad the traslatio group of V, edowed with the product measure µ of ν ad a traslatio ivariat Haar
25 Itegral Geometry ad Algebraic Structures for Tesor Valuatios 83 measure o V, ad where g is the rotatioal part of ḡ. (Agai this otatio is cosistet with the special case where Ḡ = G is the motio group, G = O( ad µ is the motio ivariat Haar measure with its usual ormalizatio as a product measure. Choosig bases ad arguig as above, we fid s G 1,s (Φ(K, L = di Φ Φ i (K Ψ (L (5 i, for K, L K (V. Of course, the costats d Φ i deped o the chose bases ad o Φ, but the operator, called itersectioal iematic operator, s G 1,s : TVal s 1+s,G (V TVal s1,g (V TVal s,g (V Φ di Φ Φ i Ψ, i, is idepedet of these choices. I the followig, we explai the coectio betwee these operators ad the provide explicit examples. For this we first lift the Poicaré duality maps to tesorvalued valuatios. Let V be a Euclidea vector space with scalar product,. For s r we defie the cotractio map by cotr : V s V r V (r s, (v 1... v s, w 1... w r v 1, w 1 v, w... v s, w s w s+1 w r, ad liearity. This map restricts to a map cotr : Sym s V Sym r V Sym r s V. I particular, if r = s, the map Sym s V Sym s V R is the usual scalar product o Sym s V, which will also be deoted by,. The trace map tr : Sym s V Sym s V is defied by restrictio of the map V s V (s, v 1... v s v 1, v v 3... v s, for s. The scalar product o Sym s V iduces a isomorphism q s : Sym s V (Sym s V ad we set pd s c : TVals, = Val Sym s V pd c qs (Val (Sym s V = (TVal s,, pd s m : TVals, = Val Sym s V pd m qs (Val (Sym s V = (TVal s,. From Propositio.7 it follows easily that Fially, we write pd s m = ( 1s pd s c. (6 m, c : TVal s 1, (V TVal s, (V TVal s 1+s, (V
26 84 Adreas Berig ad Daiel Hug for the maps correspodig to the product ad the covolutio. Moreover, we write m G, c G for the restrictios of these maps to the correspodig spaces of Gcovariat tesor valuatios. Theorem.10. Let G be a compact subgroup of O( actig trasitively o the uit sphere. The the diagram TVal s 1+s,G a G s 1,s TVal s 1,G TVal s,g pd s 1 +s c pd s 1 c pd s c ( TVal s 1 +s,g c G ( TVal s 1,G ( TVal s,g F F F ( TVal s 1 +s,g m G ( TVal s 1,G ( TVal s,g pd s 1 +s m TVal s 1+s,G G s 1,s pd s 1 m pd s m TVal s 1,G TVal s,g commutes ad ecodes the relatios betwee product, covolutio, AleserFourier trasform, itersectioal ad additive iematic formulas. This diagram allows us to express the additive iematic operator i terms of the itersectioal iematic operator, ad vice versa, with the Fourier trasform as the li betwee these operators. Corollary.11. Itersectioal ad additive iematic formulas are related by the AleserFourier trasform i the followig way: a G = ( F 1 F 1 G F, or equivaletly G = (F F a G F 1. This follows by looig at the outer square i Theorem.10, by carefully taig ito accout the sigs comig from (6. 4. Some Explicit Examples of Kiematic Formulas We start with a descriptio of a Crofto formula for tesor valuatios. Combiig the coectio betwee Crofto formulas ad the product of valuatios (see [4, ( ad (16] ad the explicit formulas for the product of tesor valuatios give i (0, we obtai
27 Itegral Geometry ad Algebraic Structures for Tesor Valuatios 85 A(, l = Φ s (K E µ l(de = [ l ] 1 ( + l l + l a [ ] 1 ( Φ s l Φ0 l (K s m=0( 1 a m( a m 1 (4π a a! a=0,a s 1 ωs m++l ω s m+ ω l Q a Φ s a +l. After simplificatio of the ier sum by meas of Zeilberger s algorithm, we obtai the Crofto formula i the Φbasis which was obtaied i [6]. Theorem.1. If, l 0 with + l ad s N 0, the A(, l Φ s (K E µ l(de = [ l ] 1 ( + l l ( + l s =0 Γ ( l + Γ ( +s 1 Γ ( +l+s (4π! Q Φ s +l (K. The result is also true i the cases, l {0, }, if the righthad side is iterpreted properly; see the commets after [6, Theorem 3]. The same is true for the ext result. Comparig the tracefree part of this formula (or by iversio, we deduce the Crofto formula for the Ψbasis, i which the result has a particularly coveiet form. Corollary.13. If, l 0 ad + l, the A(, l Ψ s (K E µ l(de = ω s++l ω s+ ω l ( + l l + l [ ] 1 Ψ s l +l (K. Alteratively, as observed i [10], Corollary.13 ca be deduced from (1, ad the Theorem.1 ca be obtaied as a cosequece. Thus, havig ow a coveiet Crofto formula for tesor valuatios, we deduce from Hadwiger s geeral itegral geometric theorem a itersectioal iematic formula i the Ψbasis. Theorem.14. Let K, M K ad {0,..., }. The G Ψ s (K gm µ(dg = = ω s+ ω s+ κ ( [ ] 1 1 Ψ s 1 (KV + (M. Let us ow prove some more refied itersectioal iematic formulas. I priciple, we could also use Corollary.11 to fid the itersectioal iematic formulas oce we ow the additive formulas. The problem is that (15 oly gives us the value of a s1,s o the basis elemet φ s 1+s, but ot o multiples of such basis elemets with
28 86 Adreas Berig ad Daiel Hug powers of the metric tesors. However, such terms appear aturally i the Fourier trasform. We therefore use Theorem.10 with V = R ad G = O(, more precisely the lower square i the diagram. I (0 we have computed the product of two tesor valuatios. For fixed (small ras s 1, s, the formula simplifies ad ca be evaluated i a closed form. For istace, if 1, l with + l ad s 1 = s = 3, we get Φ 3 ( + 1(l + 1Γ ( +l+1 Φ3 l = π 5 ( + l + 4( + l + ( + lγ ( ( Γ l ( 3Φ+l 6 π3 + 8QΦ+l 4 π Q Φ+l π Q3 Φ+l 0. (7 Let us ext wor out the vertical arrows i the diagram of Theorem.10, that is, the Poicaré duality pd m. Agai, this is a computatio ivolvig differetial forms. The result (see [6, Corollary 5.3] is pd s m (Φs, Φs = 1 s ( ( 1s π s s! ( 4 Γ ( ( +s Γ +s Γ ( + 1. (8 We ow explai how to compute the itersectioal iematic formula O( 3,3 with this owledge. Sice Φ 1 m 0, it is clear that there is a formula of the form O( 3,3 (Φ6 i = a,i, Φ 3 Φ3 l +l=+i with some costats a,i, which remai to be determied. Fix, l with + l = + i. Usig (8, we fid pd 3 m Φ3, Φ3 = 1 ( ( 7π 3 Γ ( ( +3 Γ +3 Γ ( + 1, ad therefore pd 3 m Φ3 l, Φ3 l = 1 7π 3 ( l l( l Γ ( l+3 (pd 3 m pd3 m O( 3,3 (Φ6 i, Φ 3 Φ3 l ( 1 = a,i, ( 7π 3 Γ ( π 3 ( l ( Γ l+3 Γ ( + 1, Γ ( +3 Γ ( + 1 l( l Γ ( l+3 ( Γ l+3 Γ ( + 1.
29 Itegral Geometry ad Algebraic Structures for Tesor Valuatios 87 O the other had, by (7 ad (8, m O( pd6 m (Φ6 i, Φ 3 Φ3 l = pd6 m (Φ6 i, Φ 3 Φ3 l ( + 1( l + 1Γ ( i+1 = π 5 ( i + 4( i + ( iγ ( ( l Γ pd 6 m (Φ6 i, 3Φ iπ QΦ iπ 4 Q Φ iπ Q3 Φ i 0 1 ( 1(i 1Γ ( +1 (i + 1(i 1(i 3 = π 5 Γ ( ( i+1 Γ ( Γ i. From this, the explicit value of a,i, give i the theorem follows. Comparig these expressios, we fid that (i + 1(i 1(i 3 a,i, = 40Γ ( ( +1 Γ i+1 Γ ( ( Γ l ( + 1(l + 1. We summarize the result i the followig theorem. Theorem.15. Let K, M K ad i {0,..., 1}. The G (id 3 g 3 Φ 6 i (K g 1 M µ(dg (i + 1(i 1(i 3 = 40Γ ( ( +1 Γ i+1 +l=+i Γ ( Γ ( l ( + 1(l + 1 Φ3 (K Φ3 l (M. The same techique ca be applied to all bidegrees, but it seems hard to fid a closed formula which is valid simultaeously i all cases. Refereces 1. Aleser, S., Cotiuous rotatio ivariat valuatios o covex sets. A. of Math. 149 (1999, Aleser, S., Descriptio of cotiuous isometry covariat valuatios o covex sets. Geom. Dedicata 74 (1999, Aleser, S., Berig, A., Schuster, F., Harmoic aalysis of traslatio ivariat valuatios. Geom. Fuct. Aal. 1 (011, Berig, A., Algebraic itegral geometry. I: Global differetial geometry. Spriger Proc. Math. 17, Spriger, Berli (01, arxiv: Berig, A., Fu., J.H.G., Covolutio of covex valuatios. Geom. Dedicata, 13 (006, Berig, A., Hug, D., Kiematic formulas for tesor valuatios. J. Reie Agew. Math. (to appear, arxiv: Hug, D., Scheider, R., Schuster, R., Itegral geometry of tesor valuatios. Adv. i Appl. Math., 41 (008,
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