A Characterization of L p Intersection Bodies

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1 A Characterization of L p Intersection Bodies Christoph Haberl and Monika Ludwig Dedicated to Prof. Peter M. Gruber on the occasion of his sixty-fifth birthday Abstract All GL(n) covariant L p radial valuations on convex polytopes are classified for every p > 0. It is shown that for 0 < p < 1 there is a unique non-trivial such valuation with centrally symmetric images. This establishes a characterization of L p intersection bodies. 000 AMS subject classification: 5A0 (5B11, 5B45) 1 Introduction Let L R n be a star body, that is, a compact set which is star-shaped with respect to the origin and has a continuous radial function, ρ(l, u) = max{r 0 : r u L}, u S n 1. The intersection body, I L, of L is the star body whose radial function in the direction u S n 1 is equal to the (n 1)-dimensional volume of the section of L by u, the hyperplane orthogonal to u. So, for u S n 1, ρ(i L, u) = vol(l u ), where vol denotes (n 1)-dimensional volume. Intersection bodies which arise from centrally symmetric convex bodies first appeared in Busemann [5]. They are important in the theory of area in Finsler spaces. Intersection bodies of star bodies were defined and named by Lutwak [7]. The class of intersection bodies (as defined in [7]) turned out to be critical for the solution of the Busemann-Petty problem (see [7], [9], [39]) and are fundamental in geometric tomography (see e.g. [8]), in affine isoperimetric inequalities (see e.g. [19], [36]) and the geometry of Banach spaces (see e.g. [18], [37]). Valuations allow us to obtain characterizations of many important functionals and operators on convex sets by their invariance or covariance properties with respect to suitable groups of transformations (see [1], [16], [33], [34] for information on the classical theory and [1] [4], [14], [15], [0] [], [5] for some of the recent results). Here a function Z : L G, +, where L is a class of subsets of R n and G, + is an abelian semigroup, is called a valuation if whenever K, L, K L, K L L. Z K + Z L = Z(K L) + Z(K L), 1

2 In [4], intersection bodies were characterized as GL(n) covariant valuations. To state this result, we need some additional definitions. Let P0 n denote the set of convex polytopes in R n that contain the origin in their interiors and let P = {x R n : x y 1 for every y P } denote the polar body of P P0 n. We write Sn for the set of star bodies in R n. For p > 0, the L p -radial sum K + p L of K, L S n is defined by ρ(k + p L, ) p = ρ(k, ) p + ρ(l, ) p. An operator Z : P0 n Sn, + p is called trivial, if it is a linear combination with respect to + p of the identity and central reflection. An operator Z is called GL(n) covariant of weight q, q R, if for all φ GL(n) and all bodies Q, Z(φQ) = det φ q φ Z Q, where det φ is the determinant of φ. An operator Z is called GL(n) covariant, if Z is GL(n) covariant of weight q for some q R. Theorem ([4]). An operator Z : P n 0 Sn, + 1 is a non-trivial GL(n) covariant valuation if and only if there is a constant c 0 such that for every P P n 0. Z P = c I P This theorem establishes a classification of GL(n) covariant valuations within the dual Brunn-Minkowski theory. Intersection bodies were introduced in [7] as an analogue in this dual theory of the classical projection bodies in the Brunn- Minkowski theory. In recent years, the Brunn-Minkowski theory was extended using Firey s L p Minkowski addition (see [8], [9]). In particular, Lutwak, Yang, and Zhang introduced L p projection bodies and obtained important affine isoperimetric inequalities (see [30], [31]). In [3], a valuation theoretic characterization of L p projection bodies was obtained. Here we ask the corresponding question within the dual Brunn-Minkowski theory. The notion corresponding to L p Minkowski addition is L p radial addition. So we ask for a classification of L p radial valuations. A complete answer for the planar case is given in Theorem 3 in Section 3.3. For n 3, we obtain the following result. Theorem 1. For 0 < p < 1, an operator Z : P n 0 Sn, + p is a non-trivial GL(n) covariant valuation if and only if there are constants c 1, c 0 such that Z P = c 1 I + p P + p c I p P for every P P n 0. For p > 1, all GL(n) covariant valuations Z : Pn 0 Sn, + p are trivial. Here, for Q P0 n, the star body I+ p Q is defined for u S n 1 by ρ(i + p Q, u) p = u x p dx, Q u + where u + = {x R n : u x 0}. We define I p Q = I + p ( Q).

3 As a consequence, we obtain the following characterization of L p intersection bodies. For p < 1, we call the centrally symmetric star body I p Q = I + p Q + p I p Q the L p intersection body of Q P0 n. So, for u Sn 1, ρ(i p Q, u) p = u x p dx. (1) Since vol(q u ε ) = lim u x 1+ε dx ε 0+ Q (cf. [18], p. 9), 1 p ρ(i Q, u) = lim ρ(i p Q, u) p, p 1 that is, the intersection body of Q is obtained as a limit of L p intersection bodies of Q. Also note that a change to polar coordinates in (1) shows that up to a normalization factor ρ(i p Q, u) p equals the L p cosine transform of ρ(q, ) n p. We denote by Sc n the set of centrally symmetric star bodies in R n and classify GL(n) covariant valuations Z : P0 n Sn c, + p. The planar case is contained in Theorem 4 in Section 3.3. For n 3, we obtain the following result. Theorem. For 0 < p < 1, an operator Z : P n 0 Sn c, + p is a non-trivial GL(n) covariant valuation if and only if there is a constant c 0 such that Q Z P = c I p P for every P P n 0. For p > 1, all GL(n) covariant valuations Z : Pn 0 Sn c, + p are trivial. Up to multiplication with a suitable power of the volume of Q, the L p intersection body of Q is just the polar L q centroid body of Q. For q > 1, L q centroid bodies were introduced by Lutwak and Zhang [3]. They led to important affine isoperimetric inequalities (see [6], [10], [30], [3]). Yaskin and Yaskina [38] introduced polar L q centroid bodies for 1 < q < 1 and solved the corresponding Busemann-Petty problem. For applications connected with embeddings in L q spaces, see [13], [17], and [35]. For a detailed discussion of the operators I + p and I p, we refer to [11]. Notation and Preliminaries We work in n-dimensional Euclidean space R n and write x = (x 1, x,..., x n ) for vectors x R n. The standard basis in R n will be denoted by e 1, e,..., e n. We use x y to denote the usual scalar product x 1 y 1 + x y + + x n y n of two vectors x, y R n, and to define the norm x = x x. The unit sphere {x R n : x = 1} is denoted by S n 1. Given A, A 1, A,..., A k R n, we write [A 1, A,..., A k ] for the convex hull of A 1, A,..., A k, we write lin A for the linear hull of A, and set A = {x R n : x y = 0 for all y A}. For L S n, we extend the radial function to a homogeneous function defined on R n \{0} by ρ(l, x) = x 1 ρ(l, x/ x ). Then it follows immediately from the definition that ρ(φl, x) = ρ(l, φ 1 x), x R n \{0} () for φ GL(n). 3

4 We call a valuation Z : P0 n Sn, + p an L p radial valuation. A valuation Z with Z P = {0} for every P having dimension less than n is called simple. A valuation is called GL(n) contravariant of weight q, q R, if Z φp = det φ q φ t Z P for every φ GL(n) and every P P n 0. Here φ t denotes the transpose of the inverse of φ. For 0 < p < 1, the operators I ± p : P n 0 Sn and I p : P n 0 Sn are L p radial valuations and GL(n) contravariant operators of weight 1/p. The following lemma guarantees that a classification of all L p radial valuations which are GL(n) covariant with negative weight follows from a classification of all L p radial valuations which are GL(n) contravariant with positive weight. Moreover, if we know all L p radial valuations, GL(n) covariant of arbitrary weight, we will know all GL(n) contravariant L p radial valuations and vice versa. Lemma 1. Let Z be an L p radial valuation and define another L p radial valuation Z by Z P = Z P for every P P0 n. Then Z is GL(n) covariant of weight q if and only if Z is GL(n) contravariant of weight q. Proof. That Z satisfies the valuation property is a consequence of (P Q) = P Q, (P Q) = P Q for polytopes P, Q P0 n having convex union (see, for example, [36]). The statement of the lemma follows from the fact that (φp ) = φ t P holds for every P P0 n and every φ GL(n)..1 Extension Given an L p radial valuation Z, we define another valuation Y : P0 n C +(S n 1 ) by Y P ( ) = ρ(z P, ) p. Here C + (S n 1 ) is the set of non-negative continuous functions on the sphere. We want to extend this valuation to the set P n 0 of convex polytopes which are either in P0 n or are intersections of a polytope in Pn 0 and a polyhedral cone with at most n facets having its apex at the origin. The following preparations will show when such extensions exist. For 1 j n, let P n j denote the set of polytopes which are intersections of polytopes in P0 n and j halfspaces bounded by hyperplanes H 1,..., H j containing the origin and having linearly independent normals. We need some more notation. For a hyperplane H R n, P0 n (H) is the set of convex polytopes in H containing the origin in their interiors relative to H. Let C + (S n 1 ) denote the superset of C + (S n 1 ) consisting of all non-negative functions defined almost everywhere (with respect to spherical Lebesgue measure) on S n 1 which are continuous almost everywhere. We write H +, H for the closed halfspaces bounded by H. For P P0 n(h) and A Sn 1, we say that Y : P0 n C +(S n 1 ) vanishes on A at P if for u H \H, v H + \H and every w A, there exists a neighbourhood A(w) of w such that lim Y[P, u, v] = 0 u,v 0 uniformly on A(w) holds. If there exists a constant c R such that Y[P, u, v] c for u, v 1, u H \H, v H + \H and [P, u, v] = [P, u] [P, v], then we say that Y : P0 n C + (S n 1 ) is bounded at P. 4

5 Now we are able to formulate the following lemma proved in [4]. Lemma. Let Y : P n 0 C +(S n 1 ) be a valuation. 1. If Y vanishes on S n 1 at P for every hyperplane H containing the origin and every P P0 n(h), then Y can be extended to a simple valuation Y : Pn 0 C + (S n 1 ).. If Y is bounded and vanishes on S n 1 \H at P for every hyperplane H containing the origin and every P P0 n (H), then Y can be extended to a simple valuation Y : P n 0 C + (S n 1 ) and for P P n 0 bounded by hyperplanes H 1, H,..., H n containing the origin, Y P is continuous and bounded on S n 1 \(H 1 H n ). 3. If Y is bounded and vanishes on S n 1 \H at P for every hyperplane H containing the origin and every P P0 n (H), then Y can be extended to a simple valuation Y : P n 0 C + (S n 1 ) and for P P n 0 bounded by hyperplanes H 1, H,..., H n containing the origin, Y P is continuous and bounded on S n 1 \(H1 H n ). 4. If Y vanishes on S n 1 \H at P for every hyperplane H containing the origin and every P P0 n(h), then Y can be extended to a simple valuation Y : Pn 0 C + (S n 1 ) and for P P n 0 bounded by hyperplanes H 1, H,..., H n containing the origin, Y P is continuous on S n 1 \(H1 H n ). The extension is defined inductively for j = 1,..., n, and convex polytopes P = P 0 H 1 + H+ j with P 0 P0 n and hyperplanes having linearly independent normals: For u H 1 H j 1, u Hj \H, set Y P = lim u 0 Y[P, u] on S n 1, S n 1 \(H 1 H j ) or S n 1 \(H 1 H j ) if Y vanishes on Sn 1, S n 1 \H or S n 1 \H, respectively. The proof of the following lemma is omitted since it is nearly the same as the proof of Lemma 5 and Lemma 8 in [4]. Lemma 3. Let Z : P n 0 Sn be an L p radial valuation and define Y : P n 0 C + (S n 1 ) by Y P ( ) = ρ(z P, ) p. 1. If Z is GL(n) covariant of weight q, then for every hyperplane H containing the origin and every P P 0 (H), the following holds: If q = 0, then Y vanishes on S n 1 \H at P and if q > 0, then Y vanishes on S n 1 at P. In both cases, Y is bounded at P.. If Z is GL(n) contravariant of weight q, then for every hyperplane H containing the origin and every P P 0 (H), the following holds: If q > 0, then Y vanishes on S n 1 \H at P and if q > 1, then Y vanishes on S n 1 at P. For q 1, Y is bounded at P. 5

6 Let Z be an L p radial valuation which is GL(n) contravariant of weight q. For q > 0, Lemma and Lemma 3 guarantee the existence of an extension of Y P ( ) = ρ(z P, ) p to P n 0 for which we write Y. We extend these functions from S n 1 to R n \{0} by making them homogeneous of degree p. From the definition of this extension it follows for φ GL(n) and P P n 0 bounded by hyperplanes H 1, H,..., H j that Y φp (x) = det φ pq Y P (φ t x) (3) on S n 1 \φ t (H 1 H j ) for 0 < q 1 and on Sn 1 for q > 1. If Z is an L p radial valuation which is GL(n) covariant of weight q, we proceed as above. For q 0, Lemma and Lemma 3 guarantee the existence of an extension of Y P ( ) = ρ(z P, ) p to P n 0 for which we write Y and which we extend from S n 1 to R n \{0} by making it homogeneous of degree p. From the definition of this extension it follows for φ GL(n) and P P n 0 bounded by hyperplanes H 1, H,..., H j that Y φp (x) = det φ pq Y P (φ 1 x) (4) on S n 1 \φ(h 1 H j ) for q = 0 and on S n 1 for q > 0. 3 Proof of the Classification Results We first establish a classification of valuations which are GL(n) contravariant of weight q > 0 and then a classification of valuations which are GL(n) covariant of weight q 0. By Lemma 1, combining these results gives a classification of GL(n) covariant valuations. The classification result for n 3 is contained in Theorem 1. The result for n = is stated in Section 3.3. Let Z : P0 n Sn, + p be a valuation which is GL(n) contravariant of weight q > 0 and let Z denote its extension to P n 0. Let T n = [0, e 1,..., e n ] be the standard simplex in R n. First, we show that Z is determined on P n 0 by its value on T n. Since Z is a simple valuation on P n 0, it suffices to show the statement for a polytope P P n 0 contained in a simplicial cone C bounded by n hyperplanes containing the origin and with linearly independent normal vectors. We dissect P = T 1 T k, where T i P n 0 are n-dimensional simplices with pairwise disjoint interiors. Let H be a suitable affine hyperplane such that D = C H and S i = T i H are (n 1)- dimensional simplices. We need the following notions (see [6]). A finite set of (n 1)-dimensional simplices αd is called a triangulation of D if the simplices have pairwise disjoint interiors and their union equals D. An elementary move applied to αd is one of the two following operations: a simplex S αd is dissected into two (n 1)-dimensional simplices S 1, S by an (n )-dimensional plane containing an (n 3)-dimensional face of S; or the reverse, that is, two simplices S 1, S αd are replaced by S = S 1 S if S is again a simplex. It is shown in [6] that for every triangulation αd there are finitely many elementary moves that transform αd into the trivial triangulation {D}. Note that to each (n 1)-dimensional simplex S αd, there corresponds a polytope Q P n 0 such that Q H = S. If S is dissected by an (n )-dimensional plane E H corresponding to an elementary move into S 1, S, then Q is dissected by the cone generated by E into Q 1, Q P n 0. Since Z is a simple valuation on P n 0, we obtain Z Q = Z Q 1 + p Z Q. The same 6

7 argument applies for the reverse move. Thus, after finitely many steps, we obtain that Z P = Z T 1 + p + p Z T k. Since Z is GL(n) contravariant, this proves that Z is determined on P n 0 by Z T n. We set f(x) = ρ(z T n, x) p almost everywhere on R n. Since Z is GL(n) contravariant and T n does not change when the coordinates are permutated, we obtain f(x 1,..., x n ) = f(x k1,..., x kn ) (5) for every permutation (k 1,..., k n ) of (1,..., n). We derive a family of functional equations for f. For 0 < λ j < 1 and j =, 3,..., n, we define two families of linear maps by Note that φ j e j = λ j e j + (1 λ j )e 1, φ j e k = e k for k j, ψ j e 1 = λ j e j + (1 λ j )e 1, ψ j e k = e k for k 1. φ 1 j e j = 1 e j 1 λ j e 1, φ 1 j e k = e k for k j, λ j λ j ψ 1 j e 1 = λ j 1 λ j e j λ j e 1, ψ 1 j e k = e k for k 1. Let H j be the hyperplane through 0 with normal vector λ j e 1 (1 λ j )e j. Then we have T n H + j = φ j T n and T n H j = ψ j T n. Since Z is a simple valuation, it follows that Z T n = Z(φ j T n ) + p Z(ψ j T n ). Since Z is GL(n) contravariant, this and (3) imply f(x) = λ pq j f(φt jx) + (1 λ j ) pq f(ψ t jx) (6) almost everywhere on R n where the set of exception depends on the value of q. Similar observations can be made if the valuation Z is GL(n) covariant of weight q 0. Then we have by (4) f(x) = λ pq j f(φ 1 j x) + (1 λ j ) pq f(ψj 1 x) (7) almost everywhere. Note that (5) holds in the covariant case, too. 3.1 The -dimensional Contravariant Case Lemma 4. Let Z : P 0 S, + p be a valuation which is GL() contravariant of weight q = 1. Then there exists a constant c 0 such that Z P = c ψ π (P + p ( P )) for every P P 0, where ψ π denotes the rotation by an angle π. 7

8 Proof. Since ρ(p Q, ) = max{ρ(p, ), ρ(q, )}, ρ(p Q, ) = min{ρ(p, ), ρ(q, )}, () implies that the function P cψ π (P + p ( P )) is in fact an L p radial valuation. Since ψ π π = (det φ)φ t (8) φψ 1 holds for every φ GL(), we obtain by using () ρ(cψ π (φp + p ( φp )), x) p = c p ρ((det φ)p, ψ 1 π φ t x) p + c p ρ( (det φ)p, ψ 1 π φ t x) p This proves the contravariance of weight 1. From Lemma, Lemma 3, and (6) we know that = c p ρ( det φ P, ψ 1 π φ t x) p + c p ρ( det φ P, ψ 1 π φ t x) p = ρ( det φ φ t cψ π (P + p ( P )), x) p. f(x) = λ p f(φt x) + (1 λ ) p f(ψ t x) (9) holds for every x R which does not lie in the linear hull of e 1, e or λ e 1 (1 λ )e. Thus it follows by induction that for k = 1,,..., f((ψ t )k x) = λ p k i=1 (1 λ ) p(k i) f(φ t (ψ t )i x) + (1 λ ) kp f(x) (10) holds on R except on a set consisting of countably many lines. For suitable ε > 0, we can evaluate (10) at x = e 1 εe. From this we obtain, using the homogeneity and the non-negativity of f, that f(e 1 (1 λ ) k ε(ψ t )k e ) λ p k i=1 f(φ t (e 1 (1 λ ) i ε(ψ t )i e )). (11) Note that (ψ t )k e = λ k 1 i=0 (1 λ ) i k e 1 +e. Thus e 1 (1 λ ) k ε(ψ t )k e 1. Let k in (11). By Lemma, f is uniformly bounded on S 1 \{±e 1, ±e }. So f(φ t (e 1 (1 λ ) i ε(ψ t )i e )) 0 as i. It follows from the continuity properties of f, that f((1 + ε)(e 1 + (1 λ )e )) = 0. Taking the limit ε 0, we obtain f(1, x ) = 0, for 0 < x < 1. (1) By (5), this implies f(x 1, 1) = 0, for 0 < x 1 < 1. (13) Relations (1), (13), and the homogeneity of f imply f(x 1, x ) = 0, for x 1, x > 0. (14) By evaluating (10) at e 1 εe we get in a similar way Formula (9) gives f( x 1, x ) = 0, for x 1, x > 0. (15) f( 1, 1) = λ p f( 1, 1 + λ ) + (1 λ ) p f( 1 + λ, 1). 8

9 In combination with (14) and (15) we obtain f( 1, 1) = λ p f( 1, 1 + λ ) for 1 < λ < 1, f( 1, 1) = (1 λ ) p f( 1 + λ, 1) for 0 < λ < 1. Hence c p f( 1, x ) = (1 + x ) p for 0 < x < 1, f( x 1, 1) = c p (1 + x 1 ) p for 0 < x 1 < 1, with c p = p f( 1, 1). Since f is homogeneous of degree p, we get and by (5) f( x 1, x ) = f(x 1, x ) = Combining these results finally yields almost everywhere on R. c p (x 1 + x ) p for x 1, x > 0, c p (x 1 + x ) p for x 1, x > 0. f(x) = c p ρ(ψ π T, x) p + c p ρ(ψ π ( T ), x) p For given p, q R, we define the function g p,q on R by (x pq p 1 x pq p )/(x 1 x ) pq for 0 x < x 1, g p,q (x 1, x ) = x pq p 1 /(x 1 x ) pq for x 1 > 0, x < 0, 0 otherwise. Define the linear transformations γ i, i = 0, 1,, by γ 0 (x 1, x ) = ( x 1, x ), γ 1 (x 1, x ) = (x, x 1 ), γ (x 1, x ) = ( x, x 1 ), that is, γ 0, γ 1, and γ are the reflections with respect to the origin, the first median, and the second median, respectively. Lemma 5. Let f : R \{0} R be a function positively homogeneous of degree p such that f(x 1, x ) = λ pq f(x 1, (1 λ)x 1 + λx ) + (1 λ) pq f((1 λ)x 1 + λx, x ) (16) holds on R \{0} for every 0 < λ < 1. Then f = f(1, 0) g p,q + f( 1, 0) g p,q γ 0 + f(0, 1) g p,q γ 1 + f(0, 1) g p,q γ (17) on R \{(x 1, x ) : x 1 = x }. 9

10 Proof. Equation (16) evaluated at the points ±(1, 0), ±(0, 1), ±( λ, 1 λ) and the homogeneity of f yield f(1, 1 λ) = f( 1, λ 1) = 1 (1 λ)pq p λ pq f(1, 0), (18) 1 (1 λ)pq p λ pq f( 1, 0), (19) f(λ, 1) = 1 λpq p f(0, 1), (0) (1 λ) pq f( λ, 1) = 1 λpq p f(0, 1), (1) (1 λ) pq f( λ, 1 λ) = λ pq p f( 1, 0) + (1 λ) pq p f(0, 1), () f(λ, λ 1) = λ pq p f(1, 0) + (1 λ) pq p f(0, 1). (3) First, suppose that x 1 > x 0. If x = 0, it follows from the homogeneity of f that f(x 1, 0) = x p 1 f(1, 0) = f(1, 0) g p,q(x 1, 0). For x 1 > x > 0 we obtain by (18) f(x 1, x ) = x p 1 f(1, 1 (1 x /x 1 )) = x p 1 (x /x 1 ) pq p 1 f(1, 0) (1 (x /x 1 )) pq = xpq p 1 x pq p (x 1 x ) pq f(1, 0) = f(1, 0) g p,q (x 1, x ). Since g p,q γ 0, g p,q γ 1, and g p,q γ are zero for x 1 > x 0, (17) holds in this part of the plane. (19) gives f( x 1, x ) = x p 1 f( 1, (1 x /x 1 ) 1) = x p 1 (x /x 1 ) pq p 1 f( 1, 0) (1 (x /x 1 )) pq = xpq p 1 x pq p (x 1 x ) pq f( 1, 0) = f( 1, 0)(g p,q γ 0 )( x 1, x ). But g p,q, g p,q γ 1 as well as g p,q γ vanish for x 1 < x < 0 and therefore (17) is true if x 1 < x < 0. Using the homogeneity we obtain that (17) is correct for x 1 < 0, x = 0. Now, assume x > x 1 0. If x 1 = 0, then we have f(0, x ) = x p f(0, 1) = f(0, 1)(g p,q γ 1 )(0, x ), f(0, x ) = x p f(0, 1) = f(0, 1)(g p,q γ )(0, x ). Formulae (0) and (1) for x > x 1 > 0 yield f(x 1, x ) = x p f(x 1/x, 1) = x p 1 (x 1 /x ) pq p f(0, 1) (1 (x 1 /x )) pq = xpq p x pq p 1 (x x 1 ) pq f(0, 1) = f(0, 1)(g p,q γ 1 )(x 1, x ), f( x 1, x ) = x p f( x 1/x, 1) = x p 1 (x 1 /x ) pq p f(0, 1) (1 (x 1 /x )) pq = xpq p x pq p 1 (x x 1 ) pq f(0, 1) = f(0, 1)(g p,q γ )( x 1, x ). 10

11 Since g p,q, g p,q γ 0, g p,q γ are zero for x > x 1 0 and g p,q, g p,q γ 0, g p,q γ 1 vanish for x < x 1 0, it remains to prove identity (17) if the coordinates have different signs. Finally, let x 1 and x be greater than zero. By () and (3) we have f( x 1, x ) = (x 1 + x ) p f( x 1 /(x 1 + x ), 1 x 1 /(x 1 + x )) x pq p xpq p 1 = f(0, 1) + f( 1, 0) (x 1 + x ) pq (x 1 + x ) pq = f(0, 1)(g p,q γ 1 )( x 1, x ) + f( 1, 0)(g p,q γ 0 )( x 1, x ), f(x 1, x ) = (x 1 + x ) p f(x 1 /(x 1 + x ), x 1 /(x 1 + x ) 1) x pq p xpq p 1 = f(0, 1) + f(1, 0) (x 1 + x ) pq (x 1 + x ) pq = f(0, 1)(g p,q γ )(x 1, x ) + f(1, 0)g p,q (x 1, x ). The fact that g p,q and g p,q γ are zero in the second quadrant and g p,q γ 0, g p,q γ 1 are zero in the fourth quadrant completes the proof. In the following, we have q > 1. Therefore Lemma and Lemma 3 imply that f is continuous on S n 1. Thus (6) holds on R\{0} and f satisfies the conditions of Lemma 5. Combined with (5) this implies that on R \{0}. f = f(1, 0)(g p,q + g p,q γ 1 ) + f( 1, 0)(g p,q γ 0 + g p,q γ ) (4) Lemma 6. Let Z : P 0 S, + p be a valuation which is GL() contravariant of weight q. Let p > 1, q > 1 or 0 < p < 1, q > 1/p. Then for every P P 0. Z P = {0} Proof. For x > 0 fixed, we obtain by (4) that lim f(x 1, x ) = x 1 x + lim x 1 x + x pq p 1 x pq p (x 1 x ) pq f(1, 0) has to be finite. This implies that f(1, 0) has to be zero. Considering lim x1 x + f( x 1, x ) proves f( 1, 0) = 0. So by (4), f vanishes on R \{0}. Lemma 7. For p < 1, let Z : P0 S, + p be a valuation which is GL() contravariant of weight q = 1/p. Then there exist constants c 1, c 0 such that for every P P 0. Z P = c 1 I + p P + p c I p P 11

12 Proof. A simple calculation shows almost everywhere. Therefore ρ(i + p T, ) p = (p 3p + ) 1 (g p,1/p + g p,1/p γ 1 ) (5) ρ(i p T, ) p = ρ(i + p T, γ 0 ( )) p = (p 3p + ) 1 (g p,1/p γ 0 + g p,1/p γ ). (6) Combined with (4), these equations complete the proof. Finally, we consider the case p < 1 and q (1, 1/p). We define ρ(i + p,q T, x) p = (g p,q + g p,q γ 1 )(x). The restrictions on q show that ρ(i + p,q T, ) is continuous and non-negative on R \{0}. By definition, ρ(i + p,q T, ) is positively homogeneous of degree 1 and thus is the radial function of a star body. We extend this definition to all simplices in R having one vertex at the origin (we denote this set by T0 ): { det φ I + p,q S = q φ t I + p,q T if S is -dimensional and S = φt, {0} otherwise. Note that I + p,q is well defined on T0 since ρ(i+ p,q T, ) does not change if the coordinates are interchanged. We claim that I + p,q is a valuation on T0. To prove this, it suffices to check the valuation property if the two involved simplices coincide in an edge. Since by definition I + p,q is GL() contravariant, it suffices to check the valuation property for the standard simplex. Thus it suffices to show that I + p,q T = I + p,q(t H + ) + p I + p,q(t H ) where H is the line with normal vector λe 1 (1 λ)e, 0 < λ < 1. Therefore we have to prove ρ(i + p,q T, (x 1, x )) p = λ pq ρ(i + p,q T, (x 1, (1 λ)x 1 + λx )) p +(1 λ) pq ρ(i + p,q T, ((1 λ)x 1 + λx, x )) p. (7) The case x 1, x < 0 is trivial. So assume x 1 > x 0. Then x 1 > (1 λ)x 1 +λx 0, (1 λ)x 1 + λx > x 0, and the right hand side of (7) equals pq xpq p 1 ((1 λ)x 1 + λx ) pq p λ (x 1 (1 λ)x 1 λx ) pq + (1 λ) pq ((1 λ)x 1 + λx ) pq p x pq p ((1 λ)x 1 + λx x ) pq which is nothing else than ρ(i + p,q T, (x 1, x )) p. Similar, we obtain (7) for points x > x 1 0. To check (7) for (x 1, x ), x 1, x > 0, we first assume that (1 λ)x 1 λx > 0. Then 0 < (1 λ)x 1 λx < x 1 and the sum appearing in (7) equals pq xpq p 1 ((1 λ)x 1 λx ) pq p λ (x 1 (1 λ)x 1 + λx ) pq + (1 λ) pq ((1 λ)x 1 λx ) pq p ((1 λ)x 1 λx + x ) pq. 1

13 If (1 λ)x 1 λx < 0, the right hand side of (7) is λ pq x pq p 1 (x 1 (1 λ)x 1 + λx ) pq. These two expressions are equal to ρ(i + p,q T, (x 1, x )) p. The case (1 λ)x 1 λx = 0 is simple and the remaining part can be treated in an analogous way. Now, we extend the valuation I + p,q to P 0 by setting ρ(i + p,q P, x) p = i I ρ(i + p,q S i, x) p, where {S i : i I, dim S i = } T 0 is a dissection of P, that is, I is finite, P = i I S i and no pair of simplices intersects in a set of dimension. Given two different dissections, it is always possible to obtain one from the other by a finite number of the following operations: a simplex is dissected into two - dimensional simplices by a line through the origin, or the converse, that is, two simplices whose union is again a simplex are replaced by their union (We remark that the corresponding result holds true for n 3, see [6]). Since I + p,q is a valuation on T0, this shows that I+ p,q is well defined on P 0. We have to prove that I + p,q is a valuation. To do so, let P, Q P 0 be two - dimensional convex polytopes such that their union is again convex. We dissect R into -dimensional convex cones with apex 0 in such a way that each vertex of P, Q, P Q, P Q lies on the boundary of some cone in this dissection. The intersections of such a cone with the boundary of P and Q are line segments which are either identical, do not intersect, or intersect in their endpoints only. Therefore I + p,q is a valuation and obviously it is GL() contravariant of weight q. We define the L p radial valuation I p,q by setting I p,q P = I + p,q( P ). Now, (4) implies the following result. Lemma 8. For p < 1, let Z : P0 S, + p be a valuation which is GL() contravariant of weight 1 < q < 1/p. Then there exist constants c 1, c 0 such that Z P = c 1 I + p,q P + p c I p,q P for every P P The -dimensional Covariant Case Let Z : P0 S, + p be a valuation which is GL() covariant of weight q. Let q > 0. As before, let Y P ( ) = ρ(z P, ) p and denote the extension of Y to P 0 by Y. Note that Lemma and Lemma 3 imply that Y is continuous on S 1. We define the valuation Ŷ by Ŷ P ( ) = Y P (ψ 1 π ( )) for every P P 0. From (4) and (8) it follows that for φ GL() with det φ > 0 Ŷ φp (x) = det φ pq Y P (φ 1 ψ 1 π x) = det φ pq+p Ŷ P (φ t x) 13

14 for every P P 0. So Ŷ T satisfies (16) with q + 1 instead of q. From the GL() covariance it follows that Ŷ T (x 1, x ) = Ŷ T ( x, x 1 ). Thus Lemma 5 shows that Ŷ T = Ŷ T (1, 0)(g p,q+1 + g p,q+1 γ ) + Ŷ T (0, 1)(g p,q+1 γ 0 + g p,q+1 γ 1 ). (8) Considering the limit x pq 1 lim xpq x 1 x + (x 1 x ) pq+p for fixed x > 0, we derive for p > 1 that Ŷ T (1, 0) = Ŷ T (0, 1) = 0 since Ŷ T is continuous on R \{0} and has to be finite on the first median. This limit also proves that Ŷ T (1, 0) = Ŷ T (0, 1) = 0 for p < 1 and q > 1/p 1. Now, (8) implies the following result. Lemma 9. Let Z : P 0 S, + p be a valuation which is GL() covariant of weight q > 0. Let p > 1, q > 0 or 0 < p < 1, q > 1/p 1. Then for every P P 0. Z P = {0} For p < 1 and q (0, 1/p 1), we define J + p,q by ρ(j + p,q T, x) p = (g p,q+1 + g p,q+1 γ )(ψ π x). Similar to the contravariant case, J + p,q can be extended to a covariant valuation on P 0. We define J p,q by J p,q P = J + p,q( P ). Now, (8) implies the following result. Lemma 10. For p < 1, let Z : P0 S, + p be a valuation which is GL() covariant of weight 0 < q < 1/p 1. Then there exist constants c 1, c 0 such that for every P P 0. Z P = c 1 J + p,q P + p c J p,q P For q = 1/p 1, the continuity of Ŷ T at the first median and (8) yield that Ŷ T (1, 0) = Ŷ T (0, 1). Therefore we obtain the following lemma by using (5), (6) and the identity ψ π I p P = ψ 1 π I p P. Lemma 11. Let Z : P0 S, + p be a valuation which is GL() covariant of weight q = 1/p 1. Then there exists a constant c 0 such that for every P P 0. Z P = c ψ π I p P 14

15 3.3 The -dimensional Classification Theorems Using the lemmas of the preceding sections and the planar case of Lemma 1 and Lemma 17, we obtain the following result. Theorem 3. For 0 < p < 1, an operator Z : P0 S, + p is a non-trivial valuation which is GL() covariant of weight q if and only if there are constants c 1, c 0 such that c 1 I + p P + p c I p P for q = 1/p c 1 I + p,q P + p c I p,q P for 1/p < q < 1 Z P = c 1 ψ π (P + p ( P )) for q = 1 c 1 J + p,q P + p c J p,q P for 0 < q < 1/p 1 c 1 ψ π I p P for q = 1/p 1 for every P P 0. For p > 1, an operator Z : P 0 S, + p is a non-trivial GL() covariant valuation if and only if there is a constant c 0 such that for every P P 0. Z P = c ψ π (P + p ( P )) Next, we consider an operator Z with centrally symmetric images. Note that in this case also the extended operator Z has centrally symmetric images. Using again the lemmas of the preceding sections and the planar case of Lemma 1 and Lemma 17, we obtain the following result. Here I p,q P = I + p,q P + p I p,q P. Theorem 4. For 0 < p < 1, an operator Z : P0 S c, + p is a non-trivial valuation which is GL() covariant of weight q if and only if there is a constant c 0 such that c I p P for q = 1/p c I p,q P for 1/p < q < 1 Z P = c ψ π (P + p ( P )) for q = 1 c ψ π I p,q P for 0 < q < 1/p 1 c ψ π I p P for q = 1/p 1 for every P P 0. For p > 1, an operator Z : P 0 S c, + p is a non-trivial GL() covariant valuation if and only if there is a constant c 0 such that for every P P 0. Z P = c ψ π (P + p ( P )) 15

16 3.4 The Contravariant Case for n 3 Lemma 1. Let Z : P n 0 Sn, + p, n, be a valuation which is GL(n) contravariant of weight 0 < q < 1. Then for every P P n 0. Z P = {0} Proof. From (6) we deduce that for x / lin e 1 lin e n lin(λ j e 1 (1 λ j )e j ) f(x) = λ pq j f(φt jx) + (1 λ j ) pq f(ψ t jx) (9) holds. First, we want to show that f is uniformly bounded on S n 1 \{±e 1,..., ±e n }. To do so, note that since f is positive, equation (9) for j = at (x 1, 1 λ, x 3,..., x n ) and (x 1, (1 λ ), x 3,..., x n ) gives f(x 1, (1 λ )(x 1 + λ ), x 3,..., x n ) λ pq f(x 1, 1 λ, x 3,..., x n ), (30) f(x 1, (1 λ )(x 1 λ ), x 3,..., x n ) λ pq f(x 1, (1 λ ), x 3,..., x n ). (31) Let x 1 λ, x,..., x n 0 in (30). Since f is continuous on S n 1 \{±e 1,..., ±e n } and homogeneous, it is bounded in a suitable neighbourhood of λ e 1 + (1 λ )e. Thus f is also bounded in a suitable neighbourhood of λ e 1. From (5) we conclude that f is bounded in a neighbourhood of every e i, i = 1,..., n. Proceeding in an analogous way but taking the limit x 1 λ and taking (31) into account we obtain the boundedness in suitable neighbourhoods of e i, i = 1,..., n. From (9) we know that f(φ t x) = λpq f(x) + (1 λ ) pq f(ψφ t t x) for x / lin e 1 lin e n lin(e 1 +(1 λ )e ). Thus we obtain for ( 1, 1, x 3,..., x n ) by using the homogeneity and the non-negativity of f that λ pq p f( 1, 1, x 3,..., x n ) f( λ, λ, λ x 3,..., λ x n ). Since pq p < 0 and f is bounded, this yields f( 1, 1, x 3,..., x n ) = 0, x 3,..., x n R. Evaluating (9) at ( 1, 1, x 3,..., x n ) proves 0 = λ pq f( 1, λ 1, x 3,..., x n ) + (1 λ ) pq f(λ 1, 1, x 3,..., x n ). for λ 1/. Since f is non-negative, f( 1, x, x 3,..., x n ) = 0, 1 < x < 1, x 0, x 3,..., x n R, f(x 1, 1, x 3,..., x n ) = 0, 1 < x 1 < 1, x 1 0, x 3,..., x n R. Because of (3) we also have for 1 < x 1 < 1, 1 < x < 1, x 1, x 0 and arbitrary x 3,..., x n f(x1, 1, x3,..., xn) = 0, f(1, x, x 3,..., x n ) = 0. These last four equations prove that f is equal to zero almost everywhere on R n. 16

17 Lemma 13. Let Z : P n 0 Sn, + p, n 3, be a valuation which is GL(n) contravariant of weight q = 1. Then Z P = {0} for every P P n 0. Proof. By Lemma and Lemma 3, f is continuous and uniformly bounded on S n 1 except on lin e 1 lin e n. By (6), we have for j n f(x) = λ p j f(φt jx) + (1 λ j ) p f(ψ t jx) (3) on R n except on a finite union of lines. Using this repeatedly, we get n f(x) = λ p λp nf(φ t n φ t x) + λ p λp j 1 (1 λ j) p f(ψjφ t t j 1 φ t x) +(1 λ ) p f(ψ t x) This implies for k = 1,,..., j=3 λ p λp nf(φ t n φ t x) + (1 λ ) p f(ψ t x). f((ψ t )k x) λ p λp n k i=1 (1 λ ) p(k i) f(φ t n φ t (ψ t )i x) (33) except on countably many lines. Define x = x 3 e x n e n. Evaluating (33) at suitable e 1 + x and multiplying by (1 λ ) pk shows that f(e 1 + (1 λ ) k x ) λ p λp n k f(φ t n φ t (e 1 + (1 λ ) i x )). i=1 Let k. Since f is uniformly bounded and continuous at φ t n φ t e 1 = e 1 + (1 λ )e + + (1 λ n )e n, it follows that f(φ t n φ t e 1) = 0. So we get f(1, x,..., x n ) = 0, 0 < x,..., x n < 1. From (5) we obtain (using the homogeneity of f) that f(1, x,..., x n ) = 0, x,..., x k > 0, 0 < x k+1,..., x n < 1. So f(x 1,..., x n ) = 0 for x 1,..., x n > 0. Considering e 1 + x and (33) like before shows f( x 1,..., x n ) = 0 for x 1,..., x n > 0. Note that (3) for j = and arbitrary c 1 at (c, c, 1, c,..., c) proves (since p 1) that f(c, c, 1, c,..., c) = 0. Let x 1 < 0, x,..., x n > 0, and (1 λ j )x 1 +λ j x j > 0. By (3) and the fact that f vanishes at points having all coordinates greater than zero we get f(φ t n φ t 1 x) = λ p f(x) λp n except on finitely many lines. Thus we obtain λ p λ p n f( 1, c ε, c,..., c) = f(φ t n φ t ( 1, c ε, c,..., c)) = f( 1, 1 + λ (1 + c ε), 1 + λ 3 (1 + c),..., 1 + λ n (1 + c)) 17

18 for suitable ε > 0 and λ,..., λ n > 1/(1 + c ε). The continuity of f shows f( 1, x,..., x n ) = 0, 0 < x,..., x n < c. But c 1 was arbitrary, so f( 1, x,..., x n ) = 0 for x,..., x n > 0. The homogeneity yields f( x 1, x,..., x n ) = 0 for x 1, x,..., x n > 0. In conclusion, f(x 1,..., x n ) = 0 if at most one coordinate is negative. Suppose f(x 1,..., x n ) = 0 where at most 1 k < n 1 coordinates are negative. Let x be chosen such that x 1,..., x k+1 < 0 and x k+,..., x n > 0. Suppose x < x 1 < 0. Choose λ with 0 < x 1 /x < λ < 1. Then (ψ t x) 1 = (φ t ψ t x) 1 = x 1 1 λ λ 1 λ x > 0, (ψ t x) i = (φ t ψ t x) i > 0, i = k +,..., n. Since f(ψ t x) = λpq f(φt ψ t x) + (1 λ ) pq f(x) we obtain f(x) = 0. By (5) we conclude f(x) = 0 for the case x 1 < x < 0. In the following, we have q > 1. Therefore Lemma and Lemma 3 imply that f is continuous on R n \{0}. In the proof of Lemmas 14 to 16, we use the following remark. Suppose we have two functions f 1, f which are continuous on R n \{0} satisfying (5) and such that for 0 < λ j < 1, j =,..., n, f i (x) = λ pq j f i(φ t jx) + (1 λ j ) pq f i (ψ t jx) holds on R n. Further assume that these functions are equal for all points where at most two coordinates do not vanish. Then an argument similar to that at the end of the last proof shows that these functions have to be equal. Lemma 14. For p > 1, let Z : P0 n contravariant of weight q > 1. Then Sn, + p be a valuation which is GL(n) Z P = {0} for every P P n 0. Proof. Define f(x 1, x ) = f(x 1 e 1 + x e ). Then f is continuous and satisfies the conditions of Lemma 5. The proof of Lemma 6 shows f = 0. By (5) this implies that f(x i e i + x j e j ) = 0 for arbitrary 1 i, j n. Thus f vanishes on R n \{0}. Lemma 15. For p < 1, let Z : P0 n Sn, + p be a valuation which is GL(n) contravariant of weight q = 1/p. Then there exist constants c 1, c 0 such that for every P P n 0. Z P = c 1 I + p P + p c I p P Proof. For x = x 1 e 1 + x e, note that ρ(i ± p T n, x) is a multiple of ρ(i ± p T, (x 1, x )). This and an analogous argument as before proves the lemma. 18

19 Lemma 16. For p < 1, let Z : P0 n Sn, + p be a valuation which is GL(n) contravariant of weight q > 1, q 1/p. Then for every P P n 0. Proof. By (6) we have Z P = {0} f(x) = λ pq f(φ t x) + (1 λ) pq f(ψ t x) on R n \{0}. Since e 3 is an eigenvector of φ t and ψt with eigenvalue 1, we get f(±e i) = 0 for i = 1,..., n. For f(x 1, x ) = f(x 1 e 1 +x e ) this implies f(1, 0) = f( 1, 0) = 0. Lemma 5 proves f = The Covariant Case for n 3 Lemma 17. Let Z : P n 0 Sn, + p, n, be a valuation which is GL(n) covariant of weight q = 0. Then there exist constants c 1, c 0 such that for every P P n 0. Z P = c 1 P + p c ( P ) Proof. By Lemma and Lemma 3, f is continuous and uniformly bounded on S n 1 \(e 1 e n ). By (7), the equation f(x) = f(φ 1 j x) + f(ψj 1 x) (34) holds for x / e 1 e n (λ j e 1 (1 λ j )e j ). Using this, we get by induction for k = 1,,..., k f(φ k x) = f(ψ 1 φi x) + f(x) (35) i=1 for x / e e n i=1 (e 1 + a i e ) and a suitable sequence (a i ). Define x = x 1 e 1 + x 3 e 3 + x 4 e x n e n where x 1, x 3, x 4,..., x n 0 and x 1 1 a i for every i. Then (35) at e e 1 + x and the non-negativity of f show f(λ k (e e 1 ) + x ) k i=1 f(ψ 1 (λi (e e 1 ) + x )). Let k. Since f is uniformly bounded, lim i f(ψ 1 (λi (e e 1 ) + x )) = 0. The continuity properties of f yield ( x1 f, λ ) x 1, x 3,..., x n = 0, for x 1, x 3,..., x n 0. (36) 1 λ 1 λ From (36) we obtain that f(x 1, x,..., x n ) = 0, for x 1, x,..., x n 0 and not all x i have the same sign. 19

20 For j =, 3,..., n and x = x 1 e 1 + x e + + x n e n where all the x i have the same sign, it follows that at least two coordinates of ψj 1 φ j x have different signs. Thus (34) gives f(φ n φ x) = f(x), n for x / e 1 e n (e 1 + k=0 k (1 λ +i )e +i ). (37) i=0 Evaluating (37) at (1,..., 1) gives from which we conclude f(1 + (1 λ ) + + (1 λ n ), λ,..., λ n ) = f(1,..., 1), f(x 1,..., x n ) = f(1,..., 1), for 0 < x,..., x n < 1, x 1 = n x x n. (38) But (37) for positive x,..., x n is nothing else than f(x 1,..., x n ) = f(x 1 + (1 λ )x + + (1 λ n )x n, λ x,..., λ n x n ). Choosing sufficiently small λ,..., λ n, we obtain by (38) f(x 1,..., x n ) = f(1,..., 1), for x 1,..., x n > 0, x 1 = n x x n. Similarly, we derive f( x 1,..., x n ) = f( 1,..., 1), for x 1,..., x n > 0, x 1 = n x x n. This shows that f(x) = c 1 ρ(t n, x) p + c ρ( T n, x) p. Lemma 18. Let Z : P n 0 Sn, + p, n 3, be a valuation which is GL(n) covariant of weight q > 0. Then Z P = {0} for every P P n 0. Proof. Since q > 0, Lemma and Lemma 3 imply that f is continuous on S n 1. By (7) we have f(x) = λ pq j f(φ 1 j x) + (1 λ j ) pq f(ψj 1 x) (39) on R n \{0}. The vector e 3 is an eigenvector with eigenvalue 1 of φ 1 and ψ 1. So for pq 1, (39) and (5) imply f(±e k ) = 0 for k = 1,,..., n. For pq = 1, (39) evaluated at e j for j > 1 yields f(e j )λ p j = f(e j (1 λ j )e 1 ). Since f(e j e 1 ) has to be finite and f is continuous, f(e j ) has to be zero. Thus also in this case f(±e k ) = 0 for k = 1,,..., n. Hence (39) gives f((1 λ j )e 1 + λ j e j ) = λ pq f(e j ) + (1 λ j ) pq f(e 1 ) = 0. 0

21 Therefore f(x 1 e 1 + e j ) = 0 for positive x 1. Using (39) again shows f( e 1 ) = λ pq j f( e 1) + (1 λ j ) pq+p f( e 1 + λ j e j ), and so f(x 1 e 1 + e j ) = 0 for x 1 1. But f(e j ) = λ pq+p j f( (1 λ j )e 1 + e j ) + (1 λ j ) pq f(e j ), which proves, together with the observations made before, that f(x 1 e 1 + e j ) = 0 for all x 1. By (5) this implies that f(e 1 + x j e j ) = 0 for all x j. The homogeneity of f shows f(x 1 e 1 + x j e j ) = 0 for all x 1, x j. Thus f vanishes on all points with at most two coordinates not equal to zero. We use induction on the number of non-vanishing coordinates. We assume that f equals zero on points with (j 1) non-vanishing coordinates. Set x = x e + + x j 1 e j 1. By (39), f((1 λ j )e 1 + λ j e j + x ) = λ pq j f(e j + x ) + (1 λ j ) pq f(e 1 + x ) = 0, which gives f(x 1 e 1 + e j + x /λ j ) = 0 for x 1 > 0. Therefore f(x 1 e 1 + e j + x ) = 0 for all x 1 > 0 and x = x e + + x j 1 e j 1. But by (39) f( e 1 + x ) = λ pq j f( e 1 + x ) + (1 λ j ) pq+p f( e 1 + λ j e j + (1 λ j )x ), f(e j + x ) = λ pq+p j f( (1 λ j )e 1 + e j + x ) + (1 λ j ) pq f(e j + λ j x ). So f(x 1 e 1 + e j + x ) = 0 for all x 1 and x = x e + + x j 1 e j 1. By (5), f(e 1 + x j e j +x ) = 0 for all x j and x = x e + +x j 1 e j 1. The homogeneity of f finally shows that f(x 1 e x j e j ) = 0 for all x 1,..., x j. Acknowledgements The authors would like to thank the referees for their careful reading and helpful remarks. The research of the first named author was carried out in part during a stay at the University Freiburg funded by the European Network PHD, MCRN He is grateful to Daniel Hug and Rolf Schneider for their warm hospitality during this stay. His research was also supported by FWF project P18308-N1. The second named author was supported in part by the European Network PHD, MCRN References [1] S. Alesker, Continuous rotation invariant valuations on convex sets, Ann. of Math. () 149 (1999), [] S. Alesker, Description of translation invariant valuations on convex sets with solution of P. McMullen s conjecture, Geom. Funct. Anal. 11 (001), [3] S. Alesker, Hard Lefschetz theorem for valuations, complex integral geometry, and unitarily invariant valuations, J. Differential Geom. 63 (003),

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24 Christoph Haberl: Institut für Diskrete Mathematik und Geometrie, Technische Universität Wien, Wiedner Hauptstraße 8-10/104, 1040 Wien, Austria address: Monika Ludwig: Institut für Diskrete Mathematik und Geometrie, Technische Universität Wien, Wiedner Hauptstraße 8-10/104, 1040 Wien, Austria address: 4