Laplace Transforms and Valuations

Laplace transforms and valuations

Jin Li1 and Dan Ma2,3

1Department of Mathematics, Shanghai University, Shanghai 200444, China, [email protected] 2Institut f¨urDiskrete Mathematik und Geometrie, Technische Universit¨at Wien, Wiedner Hauptstraße 8–10/104, 1040 Wien, Austria, [email protected] 3Department of Mathematics, Shanghai Normal University, Shanghai 200234, China, [email protected]

Abstract It is proved that the classical Laplace transform is a continuous valuation which is positively GL(n) covariant and logarithmic translation covariant. Conversely, these properties turn out to be suﬃcient to characterize this transform.

1 Introduction

Let f : [0, ∞) → R be a measurable function. The Laplace transform of f is given by Z ∞ −st Lf(s) = e f(t)dt, s ∈ R 0 whenever the integral converges. In the 18th century, Euler first considered this transform to solve second-order linear ordinary differential equations with constant coefficients. One hundred years later, Petzval and Spitzer named this transform after Laplace. Doetsch initiated systematic investigations in 1920s. The Laplace transform now is widely used for

2010 Mathematics subject classiﬁcation. 44A10, 52A20, 52B45 Key words and phrases. Laplace transform, valuation, GL(n) covariance, logarithmic translation covariance.

1 solving ordinary and partial differential equations. Therefore, it is a useful tool not only for mathematicians but also for physicists and engineers (see, for example, [7]). The Laplace transform has been generalized to the multidimensional setting in order to solve ordinary and partial differential equations in boundary value problems of several variables (see, for example, [6]). Let f be a compactly supported function that belongs to L1(Rn). The multidimensional Laplace transform of f is defined as Z −x·y n Lf(x) = f(y)e dy, x ∈ R . Rn n The Laplace transform is also considered on Kn, the set of n dimensional convex bodies n n (i.e., compact convex sets) in R . The Laplace transform of K ∈ Kn is defined by Z 1 −x·y n LK(x) = L( K )(x) = e dy, x ∈ R , K where 1K is the indicator function of K. Making use of the logarithmic version of this transform, Klartag [19] improved Bourgain’s esimate on the slicing problem (or hyperplane conjecture), which is one of the main open problems in the asymptotic theory of convex bodies. It asks whether every convex body of volume 1 has a hyperplane section through the origin whose volume is greater than a universal constant (see also [20] for more information). Noticing that both Laplace transforms are valuations, we aim at a deeper understanding on these classical integral transforms. A function z defined on a lattice (Γ, ∨, ∧) and taking values in an abelian semigroup is called a valuation if z(f ∨ g) + z(f ∧ g) = z(f) + z(g) (1.1) for all f, g ∈ Γ. A function z defined on some subset Γ0 of Γ is called a valuation on Γ0 if (1.1) holds whenever f, g, f ∨ g, f ∧ g ∈ Γ0. Valuations were a key ingredient in Dehn’s solution of Hilbert’s Third Problem in 1901. They are closely related to dissections and lie at the very heart of geometry. Here, valuations were considered on the space of convex bodies in Rn, denoted by Kn. Perhaps the most famous result is Hadwiger’s characterization theorem which classifies all continuous and rigid motion invariant real valued valuations on Kn. Klain [15] provided a shorter proof of this beautiful result based on the following characterization of the volume. Theorem 1.1 ([14, 15]). Suppose µ is a continuous rigid motion invariant and simple n n valuation on K . Then there exists c ∈ R such that µ(K) = cVn(K), for all K ∈ K . Here, Vn is the n dimensional volume. Other important later contributions can be found in [14, 18, 34, 35]. For more recent results, we refer to [1,2,8–13,16,17,22,24–26,30,32,37,38,40–42,46]. With the first result of this paper, we characterize the Laplace transform on convex bodies.

2 n n Theorem 1.2. A map Z : Kn → C(R ) is a continuous, positively GL(n) covariant and logarithmic translation covariant valuation if and only if there exists a constant c ∈ R such that ZK = cLK

n for every K ∈ Kn. Throughout this paper, without further remark, we briefly write positively GL(n) covariant as GL(n) covariant; see Section2 for definitions of GL( n) covariance and logarithmic n n n translation covariance. We call Z : Kn → C(R ) continuous if for every x ∈ R , we have ZKi(x) → ZK(x) whenever Ki → K with respected to the Hausdorff metric, where n Ki,K ∈ Kn. n Notice that LK(0) = Vn(K) holds for all K ∈ Kn. Thus, this characterization is a generalization of Theorem 1.1. Valuations are also considered on spaces of real valued functions. Here, we take the pointwise maximum and minimum as the join and meet, respectively. Since the indicator functions of convex bodies provide a one-to-one correspondence with convex bodies, valuations on function spaces are generalizations of valuations on convex bodies. Valuations on function spaces have been studied since 2010. Tsang [43] characterized real valued valuations on Lp-spaces. Kone [21] generalized this characterization to Orlicz spaces. As for valuations on Sobolev spaces, Ludwig [27,28] characterized the Fisher information matrix and the Lutwak-Yang-Zhang body. Other recent and interesting characterizations can be found in [3–5,29,33,36,44,45]. With the second result of this paper, we characterize the Laplace transform on functions based on Theorem 1.2 and the natural connection between indicator functions and convex 1 n 1 n bodies. Let Lc (R ) denote the space of compactly supported functions that belong to L (R ). 1 n n n We call z : Lc (R ) → C(R ) continuous if for every x ∈ R , we have z(fi)(x) → z(f)(x) 1 n whenever fi → f in L (R ). 1 n n Theorem 1.3. A map z : Lc (R ) → C(R ) is a continuous, positively GL(n) covariant and logarithmic translation covariant valuation if and only if there exists a continuous function h on R with the properties that h(0) = 0 (1.2) and that there exists a constant γ ≥ 0 that |h(α)| ≤ γ |α| (1.3) for all α ∈ R, such that z(f) = L(h ◦ f) 1 n for every f ∈ Lc (R ).

3 If we further assume that z is 1-homogeneous, that is, z(sf) = sz(f) for all s ∈ R and 1 n f ∈ Lc (R ), then we obtain the Laplace transform.

1 n n Corollary 1.4. A map z : Lc (R ) → C(R ) is a continuous, 1-homogeneous, positively GL(n) covariant and logarithmic translation covariant valuation if and only if there exists a constant c ∈ R such that z(f) = cLf, 1 n for every f ∈ Lc (R ).

2 Preliminaries and Notation

n Our setting is the n-dimensional Euclidean space R with the standard basis {e1, . . . , en}, where n ≥ 1. The convex hull of a set A is denoted by [A] and the convex hull of a set A and a point x ∈ Rn will be briefly written as [A, x] instead of [A, {x}]. A hyperplane is an n − 1 n n P dimensional affine space in R . The unit cube C = 1≤i≤n[o, ei] and the standard simplex n T = [o, e1, . . . , en] are two important convex bodies in this paper. The Hausdorff distance of K,L ∈ Kn is

d(K,L) = inf{ε > 0 : K ⊂ L + εB, L ⊂ K + εB}.

1 n 1 The norm on the space Lc (R ) is the ordinary L norm which is denoted by k·k. 1 n n A map z : Lc (R ) → C(R ) is called GL(n) covariant if z(f ◦ φ−1)(x) = |det φ| z(f)(φtx) (2.1)

1 n n for all f ∈ Lc (R ), φ ∈ GL(n) and x ∈ R . In this paper, we actually deal with positive GL(n) covariance, that is (2.1) is supposed to hold for all φ ∈ GL(n) that have positive 1 n n determinant. Also, a map z : Lc (R ) → C(R ) is called logarithmic translation covariant if z(f(· − t))(x) =e−x·tz(f)(x)

1 n n for all f ∈ Lc (R ) and t, x ∈ R . This deﬁnition is motivated by the relation log L(f(· − t))(x) = −x · t + log Lf(x)

(see Theorem 3.1). n n A map Z : Kn → C(R ) is called GL(n) covariant if Z(φK)(x) = |det φ| ZK(φtx)

4 n n n n for all K ∈ Kn, φ ∈ GL(n) and x ∈ R . Also, a map z : Kn → C(R ) is called logarithmic translation covariant if

Z(K + t)(x) = e−t·xZK(x)

n n for all K ∈ Kn and t, x ∈ R . Again, it is motivated by the relation log L(K + t)(x) = −t · x + log LK(x)

(see Theorem 3.3). If a valuation vanishes on lower dimensional convex bodies, we call it simple. 1 n n As we will see in Lemma 3.2, if z : Lc (R ) → C(R ) is continuous, GL(n) covariant and n n 1 logarithmic translation covariant, then Z : Kn → C(R ) deﬁned by ZK = z( K ) is also continuous, GL(n) covariant and logarithmic translation covariant, respectively. 1 n n For the constant zero function, if z : Lc (R ) → C(R ) is GL(n) covariant and logarithmic translation covariant, then

z(0) ≡ 0. (2.2)

t Indeed, z(0)(φ x) = z(0)(x) for any φ ∈ GL(n). Let x = e1. We have that z(0) is a constant function on Rn \{0}. The continuity of the function z(0) now gives that z(0) ≡ c on Rn for a constant c ∈ R. Since z is also logarithmic translation covariant, z(0)(x) = e−t·xz(0)(x) for any x, t ∈ Rn. Hence z(0) ≡ 0.

3 Laplace transforms

In this section, we study some properties of Laplace transforms.

Theorem 3.1. Let h be a continuous function on R satisfying (1.2) and (1.3). If a map 1 n n z : Lc (R ) → C(R ) satisﬁes Z z(f)(x) = (h ◦ f)(y)e−x·ydy Rn

n 1 n for every x ∈ R and f ∈ Lc (R ), then z is a continuous, GL(n) covariant and logarithmic translation covariant valuation. 1 n In particular, if h(α) = α for all α ∈ R, the Laplace transform L on Lc (R ) is a continuous, GL(n) covariant and logarithmic translation covariant valuation.

5 1 n n Proof. Let f, g ∈ Lc (R ) and E = {x ∈ R : f(x) ≤ g(x)}. Then Z z(f ∨ g)(x) = h ◦ (f ∨ g)(y)e−x·ydy n ZR Z = (h ◦ g)(y)e−x·ydy + (h ◦ f)(y)e−x·ydy E Rn\E for every x ∈ Rn. Similarly, we have Z z(f ∧ g)(x) = h ◦ (f ∧ g)(y)e−x·ydy n ZR Z = (h ◦ f)(y)e−x·ydy + (h ◦ g)(y)e−x·ydy E Rn\E for every x ∈ Rn. Thus, z(f ∨ g)(x) + z(f ∧ g)(x) Z Z = (h ◦ f)(y)e−x·ydy + (h ◦ g)(y)e−x·ydy Rn Rn =z(f)(x) + z(g)(x) for every x ∈ Rn. 1 n Next, we are going to show that z is continuous. Let f ∈ Lc (R ) and let {fi} be a sequence 1 n 1 n in Lc (R ) that converges to f in L (R ). We will show the continuity of z by showing that for every subsequence {z(f )(x)} of {z(f )(x)}, there exists a subsequence {z(f )(x)} that ij i ijk converges to z(f)(x) for every x ∈ Rn. n n Let {fij } be a subsequence of {fi} and y ∈ R . Then, for every x ∈ R , the sequence of −x·y −x·y functions y 7→ fij (y)e converges to the function y 7→ f(y)e as j → ∞ with respect to the L1 norm. It follows that there exists a subsequence {f } of {f } and a nonnegative ijk ij 1 n function Fx ∈ L (R ) such that (i) f (y)e−x·y → f(y)e−x·y almost every y with respect to Lebesgue measure; ijk (ii) |f (y)|e−x·y ≤ F (y) almost every y with respect to Lebesgue measure (see [23, Section ijk x 2.7]). Since h is continuous, we obtain

h ◦ f → h ◦ f a.e. ijk Also since h satisﬁes (1.3), we have

|h ◦ f (y)| ≤ γ|f (y)| ≤ γF (y)ex·y. ijk ijk x

6 R x·y −x·y Note that n Fx(y)e ·e dy < ∞. We conclude from the dominated convergence theorem R that Z Z −x·y −x·y z(f)(x) = (h ◦ f)(y)e dy = lim (h ◦ fi )(y)e dy = lim z(fi )(x). k→∞ jk k→∞ jk Rn Rn Moreover, for φ ∈ GL(n), Z z(f ◦ φ−1)(x) = (h ◦ f ◦ φ−1)(y)e−x·ydy n R Z = |det φ| (h ◦ f)(w)e−x·(φw)dw n ZR = |det φ| (h ◦ f)(w)e−(φtx)·wdw Rn = |det φ| z(f)(φtx)

n 1 n n for every x ∈ R and f ∈ Lc (R ). Finally, let t ∈ R . Then Z z(f(· − t))(x) = (h ◦ f)(y − t))e−x·ydy n ZR = (h ◦ f)(w)e−x·(w+t)dw n R Z =e−x·t (h ◦ f)(w)e−x·wdw Rn =e−x·tz(f)(x)

n 1 n for every x ∈ R and f ∈ Lc (R ). Next, we turn to the Laplace transform on convex bodies. 1 n n Lemma 3.2. If z : Lc (R ) → C(R ) is a continuous, GL(n) covariant and logarithmic n n translation covariant valuation, then for any α ∈ R, Z : Kn → C(R ) deﬁned by

ZK = z(α1K ) n is a continuous, GL(n) covariant and logarithmic translation covariant valuation on Kn. n Proof. For K, L, K ∪ L, K ∩ L ∈ Kn, we have

Z(K ∪ L) + Z(K ∩ L) = z(α1K∪L) + z(α1K∩L)

= z((α1K ) ∨ (α1L)) + z((α1K ) ∧ (α1L))

= z(α1K ) + z(α1L) = ZK + ZL.

7 n n Also, for a sequence {Ki} in Kn that converges to K ∈ Kn as i → ∞, we have 1 1 kα Ki − α K k → 0 as i → ∞. Indeed, for any 0 < ε ≤ 1, we have

Ki ⊂ K + εB, K ⊂ Ki + εB for suﬃciently large i. Hence (Ki \ K) ∪ (K \ Ki) ⊂ {x : ∃ y ∈ bd K, s.t. d(x, y) ≤ ε}, where bd K is the boundary of K. Hence, Z 1 1 |α Ki (y) − α K (y)|dy ≤ |α|Vn({x : ∃ y ∈ bd K, s.t. d(x, y) ≤ ε}) Rn ≤ |α| · 2εS(K + B)

1 n for suﬃciently large i. Here, S denotes the surface area. By the continuity of z on Lc (R ), we obtain 1 1 Z(Ki) = z(α Ki ) → z(α K ) = ZK n as i → ∞. Moreover, for each φ ∈ GL(n) and K ∈ Kn, we have

−1 Z(φK) = z(α1φK ) = z(α1K ◦ φ ) t t = |det φ| z(α1K ) ◦ φ = |det φ| ZK ◦ φ .

Finally, for each t, x ∈ Rn, we have

Z(K + t)(x) = z(α1K+t)(x) = z(α1K (· − t))(x) −t·x −t·x = e z(α1K )(x) = e ZK(x).

The following theorem directly follows from the deﬁnition of the Laplace transform, Theorem 3.1 and Lemma 3.2.

n Theorem 3.3. The Laplace transform on Kn is a continuous, GL(n) covariant and logarithmic translation covariant valuation.

4 Characterizations of Laplace transforms

n In this section, we ﬁrst characterize the Laplace transform on Kn as a continuous, GL(n) covariant and logarithmic translation covariant valuation. Afterwards, via an approach 1 n developed from Tsang’s in [43], we further characterize the Laplace transform on Lc (R ).

8 4.1 The Laplace transform on convex bodies We ﬁrst need to extend the valuation to Kn.

n n Lemma 4.1. If Z : Kn → C(R ) is a continuous, GL(n) covariant and logarithmic translation covariant valuation, then Z : Kn → C(Rn) deﬁned by ( ZK(x), dim K = n, ZK(x) = 0, dim K < n is a simple, GL(n) covariant and logarithmic translation covariant valuation on Kn.

Proof. The GL(n) covariance and the logarithmic translation covariance are trivial. It remains to show that

+ − n ZK(x) = Z(K ∩ H )(x) + Z(K ∩ H )(x), x ∈ R (4.1)

+ − n for every hyperplane H (when n = 1, H is a single point) such that K,K ∩H ,K ∩H ∈ Kn. Since Z is logarithmic translation covariant, we can assume w.l.o.g. that o ∈ (int K ∩ H). + We can further assume that en ⊥ H and en ∈ H due to the GL(n) covariace of Z. For a ﬁxed K, note that ±sen ∈ K for suﬃciently small s > 0. Hence the valuation property of Z shows that

+ − ZK(x) + Z[K ∩ H, ±sen](x) = Z[K ∩ H , −sen](x) + Z[K ∩ H , sen](x) (4.2) for every x ∈ Rn and suﬃciently small s > 0. The GL(n) covariance of Z gives that

Z[K ∩ H, ±sen](x) = sZ[K ∩ H, ±en](x1e1 + ··· + xn−1en−1 + sxnen),

n where x = x1e1 + ··· + xnen ∈ R . Since

lim Z[K ∩ H, ±en](x1e1 + ··· + xn−1en−1 + sxnen) s→0+

= Z[K ∩ H, ±en](x1e1 + ··· + xn−1en−1), we have

lim Z([K ∩ H, ±sen])(x) → 0. (4.3) s→0+ Now combing (4.2) and (4.3) with the continuity of Z, we get that (4.1) holds true. Next we consider ZCn.

9 Lemma 4.2. If Z : Kn → C(Rn) is a simple, GL(n) covariant and logarithmic translation covariant valuation, then there exists a constant c ∈ R such that Z n n −re1·y ZC (re1) = cLC (re1) = c e dy, (4.4) Cn for every r ∈ R. Proof. First note that Z 1 e−re1·ydy = (1 − e−r). (4.5) Cn r

For s > 0, let ψs ∈ GL(n) such that ψse1 = se1 and ψsek = ek for 2 ≤ k ≤ n. For integers p, q > 0, since Z is simple, we have

q−1 X je1 Z(ψ Cn)(e ) = Z ψ Cn + (e ). q/p 1 1/p p 1 j=0

Also since Z is GL(n) and logarithmic translation covariant, we have

q−1 q q 1 X e1 ZCn e = e−j/pZCn p p 1 p p j=0 1 e 1 − e−q/p = ZCn 1 . p p 1 − e−1/p In particular, if q = p, we have

e p(1 − e−1/p) ZCn 1 = ZCn(e ). p 1 − e−1 1

n ZC (e1) Combining the two formulas above with (4.5), and letting c = 1−e−1 ,(4.4) holds for r = q/p. Now since ZCn is a continuous function on Rn,(4.4) holds for r ≥ 0. For r < 0. Repeating the same process for −e1, we obtain q−1 q q 1 X e1 ZCn − e = ej/pZCn − p p 1 p p j=0 1 e 1 − eq/p = ZCn − 1 p p 1 − e1/p

10 and e p(1 − e1/p) ZCn − 1 = ZCn(−e ). p 1 − e 1

n 0 ZC (−e1) Combining the two formulas above and letting c = − 1−e , we have Z n 0 −re1·y ZC (re1) = c e dy Cn for r = −q/p. The continuity of the function ZCn gives that c = c0 and thus (4.4) holds for r ≤ 0.

Now we consider valuations on polytopes. Let Pn denote the set of polytopes in Rn and let Z : Pn → C(Rn) be a valuation. The inclusion-exclusion principle states that Z extends uniquely to U(P), the set of ﬁnite unions of polytopes, with

X j−1 X Z(P1 ∪ · · · ∪ Pm) = (−1) Z(Pi1 ∩ · · · ∩ Pij )

1≤j≤m 1≤i1<···

n for every P1,...,Pm ∈ P (see [31] or [39, Theorem 6.2.1 and Theorem 6.2.3]).

Lemma 4.3. If Z : Pn → C(Rn) is a simple, GL(n) covariant and logarithmic translation covariant valuation such that

n ZC (re1) = 0 (4.6) for every r ∈ R, then n ZT (re1) = 0 for every r ∈ R. Proof. The case n = 1 is trivial. We only consider n ≥ 2. First we prove that ZT n(o) = 0. Since Cn = S {x ∈ n : 0 ≤ x ≤ · · · ≤ 1≤i1<···integer n n n m ≥ 0. For integer k ≥ 1, denote Mk = kT ∩ C . Note that when k ≥ n, we have

n n Mk = C . (4.7)

11 For 1 ≤ k ≤ n − 1,

n ! n n [ n n n kT ∪ C = (kT ∩ {x ∈ R : xj ≥ 1}) ∪ C (4.8) j=1

n n Denote Tj = kT ∩ {x ∈ R : xj ≥ 1}. We have

n Tj = (k − 1)T + ej, and

n Tj1 ∩ · · · ∩ Tji = (k − i)T + ej1 + ··· + eji for i ≤ k and 1 ≤ j1 < ··· < ji ≤ n. Hence, the valuation property (after extension), inclusion-exclusion principle, simplicity and logarithmic translation covariance of Z, combined with (4.6) and (4.8), give that

n n n n Z(Mk )(se1) = Z(kT )(se1) − Z(kT ∪ C )(se1) k−1 ! n X i−1 X = Z(kT )(se1) − (−1) Z(Tj1 ∩ · · · ∩ Tji )(se1)

i=1 1≤j1<···

n−1 −s n−1 where 1 ≤ k ≤ n − 1, ai(s) = i−1 e + i for 1 ≤ i ≤ n − 1, and a0(s) = 1. For non negative integers k1, . . . , kn satisfying k = k1 + ··· + kn ≤ m − 1, we have

n n n mT ∩ (C + k1e1 + ··· + knen) = Mm−k + k1e1 + ··· + knen.

For m ≥ n, applying the valuation property, simplicity, logarithmic translation covariance of

12 Z,(4.6), (4.7) and (4.9), we have

n g(m, s) = Z(mT )(se1) m−1 X X n n = Z(mT ∩ (C + k1e1 + ··· + knen))(se1)

k=0 k1+···+kn=k, k1,...,kn≥0 m−1 X X n = Z(Mm−k + k1e1 + ··· + knen)(se1) k=0 k1+···+kn=k, k1,...,kn≥0 m−1 k X X −k1s X n = e Z(Mm−k)(se1) k=0 k1=0 k2+···+kn=k−k1, k2,...,kn≥0 m−1 k X X k − k1 + n − 2 = Z(M n )(se ) e−k1s m−k 1 n − 2 k=0 k1=0 m−1 m−k−1 ! k ! X X X k − k1 + n − 2 = (−1)ia (s)g(m − k − i, s) e−k1s i n − 2 k=m−n+1 i=0 k1=0 n−1 X = bj(m, s)g(j, s), (4.10) j=1

m−j k P m−k−j P −k1sk−k1+n−2 where bj(m, s) = (−1) am−k−j(s) e n−2 for 1 ≤ j ≤ n − 1. k=m−n+1 k1=0 Since Z is GL(n) covariant, we have

g(m, s) = mng(1, ms).

Hence the equation (4.10) gives that

n−1 X n g(1, ms) = (j/m) bj(m, s)g(1, js). (4.11) j=1

For any ﬁxed r 6= 0, taking s = r/m in (4.11), we get

n−1 X n g(1, r) = (j/m) bj(m, r/m)g(1, jr/m). (4.12) j=1

13 Since g(1, ·) is a continuous function and g(1, 0) = ZT n(o) = 0, if we can show that n (j/m) bj(m, r/m) is ﬁnite when m → ∞, then g(1, r) = 0 which gives the desired result for r 6= 0. Indeed, for suﬃciently large m,(m + n)/m ≤ 2 and ai(r/m), 1 ≤ i ≤ n − 1 are smaller than a constant N > 0. Hence, for 1 ≤ j ≤ n − 1,

m−j k X X k − k1 + n − 2 |(j/m)nb (m, r/m)| ≤ (n/m)n a (r/m) e−k1r/m j m−k−j n − 2 k=m−n+1 k1=0 m−j k n X X −k1r/m n−2 ≤ (n/m) am−k−j(r/m) e (m + n)

k=m−n+1 k1=0 m−j X |1 − e−(k+1)r/m| = (m + n)n−2(n/m)n a (r/m) m−k−j |1 − e−r/m| k=m−n+1 m−j X |1 − e−(k+1)r/m| ≤ 2n−2nnN(1/m)2 |1 − e−r/m| k=m−n+1 (n − j) max{1, e−r} ≤ 2n−2nnN(1/m)2 . |1 − e−r/m|

2 1 n Note that (1/m) |1−e−r/m| → 0 when m → ∞. Hence (j/m) bj(m, r/m) → 0 when m → ∞.

For 0 < λ < 1, let Hλ be the hyperplane through the origin with normal vector (1 − n n λ)e1 − λe2. Since Z : P → C(R ) is a simple valuation,

n n − n + n ZT (x) = Z(T ∩ Hλ )(x) + Z(T ∩ Hλ )(x), x ∈ R . (4.13)

We deﬁne φ1, φ2 ∈ GL(n) by

φ1e1 = λe1 + (1 − λ)e2, φ1e2 = e2, φ1ei = ei, for 3 ≤ i ≤ n, and φ2e1 = e1, φ2e2 = λe1 + (1 − λ)e2, φ2ei = ei, for 3 ≤ i ≤ n. n − n n + n Note that T ∩ Hλ = φ1T , T ∩ Hλ = φ2T . The GL(n) covariance of Z and valuation relation (4.13) show that

n n t n t ZT (x) = λZT (φ1x) + (1 − λ)ZT (φ2x)

14 Let f(·) = ZT n(·). We have

t t f(x) = λf(φ1x) + (1 − λ)f(φ2x) (4.14)

t n t t for every 0 < λ < 1 and x = (x1, . . . , xn) ∈ R , where φ1x = (λx1 +(1−λ)x2, x2, x3, . . . , xn) t t and φ2x = (x1, λx1 + (1 − λ)x2, x3, . . . , xn) .

Lemma 4.4. Let n ≥ 2 and let the function f ∈ C(Rn) satisfy the following properties. (i) f satisﬁes the functional equation (4.14); (ii) For every even permutation π and x ∈ Rn, f(x) = f(πx).

If f(re1) = 0 for every r ∈ R, then f(x) = 0 for every x ∈ Rn. Proof. We prove the statement by induction on the number m of coordinates of x not equal to zero. By property (ii), we can assume that the ﬁrst m coordinates of x are not equal to zero. It is trivial that the statement is true for m = 1. Assume that the statement holds true for m − 1. We want to show that

f(x1e1 + ··· + xmem) = 0 (4.15) for all the x1, . . . , xm not zero. For x1 > x2 > 0 or 0 > x2 > x1, taking x = x1e1 + x3e3 + x2 ··· + xmem, λ = in (4.14), we get x1

f(x1e1 + x3e3 + ··· + xmem) x2 x2 = f(x2e1 + x3e3 + ··· + xmem) + 1 − f(x1e1 + x2e2 + x3e3 + ··· + xmem). (4.16) x1 x1

x1 For x2 > x1 > 0 or 0 > x1 > x2, taking x = x2e2 + x3e3 + ··· + xmem, 1 − λ = in (4.14), x2 we get

f(x2e2 + x3e3 + ··· + xmem), x1 x1 = 1 − f(x1e1 + x2e2 + x3e3 + ··· + xmem) + f(x1e2 + x3e3 + ··· + xmem). (4.17) x2 x2

15 x2 For x1 > 0 > x2 or x2 > 0 > x1, taking 0 < λ = < 1 and x = x1e1 + x2e2 + x3e3 + ··· + x2−x1 xmem in (4.14), we get

f(x1e1 + x2e2 + x3e3 + ··· + xmem) x2 −x1 = f(x2e2 + x3e3 + ··· + xmem) + f(x1e1 + x3e3 + ··· + xmem). (4.18) x2 − x1 x2 − x1 Now, combined with the induction assumption and the continuity of f,(4.16), (4.17) and (4.18) show that (4.15) holds true. Proof of Theorem 1.2. By Theorem 3.3, cL is a continuous, GL(n) covariant and logarithmic n translation covariant valuation on Kn. n n Now we turn to the reverse statement. Since Z : Kn → C(R ) is a continuous, GL(n) covariant and logarithmic translation covariant valuation, Lemma 4.1 allows us to extend this valuation to a simple, GL(n) covariant and logarithmic translation covariant valuation on Kn. Hence Lemma 4.2 gives that there exists a constant c ∈ R such that

n n ZC (re1) = cLC (re1) for every r ∈ R. Now deﬁne Z0 : Pn → C(Rn) by

0 n Z P (x) = ZP (x) − cLP (x), x ∈ R . It is easy to see that Z0 is also a simple, GL(n) covariant and logarithmic translation covariant n 0 n 0 valuation on K . Also Z C (re1) = 0 for every r ∈ R. Applying Lemma 4.3 (for Z ) and Lemma 4.4 (for f = Z0T n) we get Z0T n = 0. Now using the inclusion-exclusion principle and the GL(n) covariance and the simplicity of Z0 again, we have Z0P = 0 for every P ∈ Pn since every P ∈ Pn can be dissected into ﬁnite pieces of GL(n) (with positive determinant) transforms and translations of T n. Hence

ZP = cLP

n n for every P ∈ P . Since both Z and L are continuous on Kn, ZK = cLK

n for every K ∈ Kn.

16 4.2 Laplace transforms on functions We ﬁrst consider indicator functions of Borel sets. 1 n n Lemma 4.5. If a map z : Lc (R ) → C(R ) is a continuous, GL(n) covariant and logarithmic translation covariant valuation, then there exists a continuous function h on R satisfying (1.2) and (1.3) such that Z −x·y z(α1E)(x) = h(α) 1E(y)e dy Rn for every α ∈ R, x ∈ Rn and bounded Borel set E ⊂ Rn. n n Proof. For any α ∈ R, deﬁne Zα : Kn → C(R ) by

ZαK = z(α1K ) n for every K ∈ Kn. Lemma 3.2 shows that Zα is a continuous, GL(n) and logarithmic n translation covariant valuation on Kn. Therefore, by Theorem 1.2, there exists a function h ∈ R such that

z(α1K )(x) = ZαK(x) = h(α)LK(x) Z Z −x·y −x·y = h(α) e dy = h(α) 1K (y)e dy K Rn n n for every K ∈ Kn and x ∈ R . The function h is continuous since z is continuous and kαi1K − α1K k → 0 whenever αi → α. z(0) = 0 (see (2.2)) gives that h(0) = 0. In particular, this representation holds on Par(n), the set of ﬁnite union of cubes, by the inclusion-exclusion principle. Now we consider Borel sets in Rn. For each bounded Borel set E ⊂ Rn, there exists 1 1 1 n a sequence {Ki} in Par(n) such that α Ki → α E in Lc (R ) for every α ∈ R as i → ∞. Moreover, for every x ∈ Rn, Z Z 1 1 −x·y 1 1 −x·y 0 ≤ lim ( Ki (y) − E(y)) e dy ≤ lim | Ki (y) − E(y)|e dy i→∞ i→∞ Rn Rn Z −x·y ≤ max e lim |1K (y) − 1E(y)|dy y∈E i→∞ i Rn = 0. Thus, the continuity of z gives that Z 1 1 1 −x·y z(α E)(x) = lim z(α Ki )(x) = h(α) lim Ki (y)e dy i→∞ i→∞ n Z R −x·y = h(α) 1E(y)e dy. Rn

17 The last step is to show that h satisﬁes (1.3). If h does not satisfy (1.3), then there exists a sequence {αj} in R \{0} (since h satisﬁes (1.2)) such that i |h(αj)| > 2 |αj| . (4.19)

−j n−1 Set Ej = [0, 2 /|αj|] × [0, 1] . We have Z 2−j dy = . Ej |αj| 1 1 n Let gj = αj Ej and f ≡ 0. Clearly gj, f ∈ Lc (R ). Since Z Z −j |gj(y)| dy = |αj| dy = 2 → 0 n R Ej when j → ∞. Hence kgj − fk → 0. The continuity of z now implies that

z(f)(o) = lim z(gj)(o)dy. j→∞ On the other hand, z(f)(o) = 0 (see (2.2)) and the above statement gives that Z 1 z(gj)(o) = h(αj) Ej (y)dy Rn However, since h satisﬁes (4.19), we obtain Z 0 = |z(f)(o)| = lim |z(gj)(o)| = lim |h(αj)| dy j→∞ j→∞ Ej

|h(αj)| = lim j j→∞ 2 |αj|

|h(αj)| = lim sup j j→∞ 2 |αj| ≥ 1. It is a contradiction. Hence h satisﬁes (1.3). Next, we deal with simple functions

1 n n Lemma 4.6. Let z : Lc (R ) → C(R ) be a valuation. Suppose that there exists a continuous function h on R satisfying (1.2) and (1.3) such that Z −x·y z(α1E)(x) = h(α) 1E(y)e dy Rn 18 for every α ∈ R, x ∈ Rn and bounded Borel set E ⊂ Rn. Then Z z(g) = (h ◦ g)(y)e−x·ydy Rn

1 n for every simple function g ∈ Lc (R ). 1 n Pm 1 Proof. Let g ∈ Lc (R ) be a simple function. We can write g = i=1 αi Ei , where E1,...,Em are disjoint bounded Borel sets and α1, . . . , αm ∈ R. Hence m m X 1 1 1 X 1 z(g) = z αi Ei = z((α1 E1 ) ∨ · · · ∨ (αm Em )) = z(αi Ei ), (4.20) i=1 i=1 where the last equation is the inclusion-exclusion principle for valuations on the lattice 1 n (Lc (R ), ∨, ∧). Pm 1 Since h ◦ g = i=1 h(αi) Ei , by (4.20), we obtain m m Z X 1 X 1 −x·y z(g) = z(αi Ei ) = h(αi) Ei (y)e dy n i=1 i=1 R Z m Z X 1 −x·y −x·y = h(αi) Ei (y)e dy = (h ◦ g)(y)e dy. n n R i=1 R

Finally, we are ready to prove Theorem 1.3. Proof of Theorem 1.3. Theorem 3.1 shows that f 7→ L(h◦f) is a continuous, GL(n) covariant and logarithmic translation covariant valuation. It remains to show the reverse statement. 1 n For a nonnegative function f ∈ Lc (R ), there exists an increasing sequence of nonnegative 1 n simple functions {gk} ⊂ Lc (R ) such that gk ↑ f pointwise. The monotone convergence 1 n theorem gives that kgk − fk → 0. Note that every function f ∈ Lc (R ) can be written as f = f+ − f−, where ( ( f(x), x ∈ {f ≥ 0} 0, x ∈ {f ≥ 0} f+ = , f− = . 0, x ∈ {f < 0} −f(x), x ∈ {f < 0}

Hence the above statement gives that there exists a sequence of simple functions {gk} ⊂ 1 n Lc (R ) such that gk → f pointwise and kgk − fk → 0 by the triangle inequality. Moreover,

19 n the increasing sequence |gk(x)| ↑ |f(x)| for every x ∈ R . Due to the continuity of z, Lemma 4.5 and Lemma 4.6, we have Z −x·y z(f)(x) = lim z(gk)(x) = lim (h ◦ gk)(y)e dy, (4.21) k→∞ i→∞ Rn where h is a continuous function satisfying (1.2) and (1.3). Therefore,

|h ◦ gk| ≤ γ|gk| ≤ γ|f|.

The dominated convergence theorem, the continuity of h, and (4.21) now yield Z −x·y z(f) = lim (h ◦ gk)(y)e dy i→∞ n Z R = (h ◦ f)(y)e−x·ydy Rn = L(h ◦ f).

Acknowledgement

The authors wish to thank the referee for valuable suggestions and careful reading of the original manuscript. The work of the ﬁrst author was supported in part by the National Natural Science Foundation of China (11671249) and the Shanghai Leading Academic Discipline Project (S30104). The work of the second author was supported in part by the European Research Council (ERC) Project 306445 and the Austrian Science Fund (FWF) Project P25515-N25.

References

[1] S. Alesker, Continuous rotation invariant valuations on convex sets, Ann. Math. (2) 149 (1999), no. 3, 977–1005.

[2] S. Alesker, Description of translation invariant valuations on convex sets with solution of P. McMullen’s conjecture, Geom. Funct. Anal. 11 (2001), no. 2, 244–272.

[3] Y. Baryshnikov, R. Ghrist and M. Wright, Hadwiger’s theorem for deﬁnable functions, Adv. Math. 245 (2013), 573–586.

20 [4] L. Cavallina, Non-trivial translation-invariant valuations on L∞, 2015. ArXiv:1505.00089.

[5] L. Cavallina and A. Colesanti, Monotone valuations on the space of convex functions, Anal. Geom. Metr. Spaces 3 (2015), no. 1, 167–211.

[6] J. C. Debnath, n-dimensional Laplace transforms with associated transforms and boundary value problems, Ph.D. thesis, Iowa State University, 1988.

[7] G. Doetsch, Introduction to the Theory and Application of the Laplace Transformation, Springer-Verlag, Berlin-Heidelberg, 1974.

[8] C. Haberl, Star body valued valuations, Indiana Univ. Math. J. 58 (2009), no. 5, 2253– 2276.

[9] C. Haberl, Blaschke valuations, Amer. J. Math. 133 (2011), no. 3, 717–751.

[10] C. Haberl, Minkowski valuations intertwining the special linear group, J. Eur. Math. Soc. 14 (2012), no. 5, 1565–1597.

[11] C. Haberl and M. Ludwig, A characterization of Lp intersection bodies, Int. Math. Res. Not. 10548 (2006), 1–29.

[12] C. Haberl and L. Parapatits, The centro-aﬃne Hadwiger theorem, J. Amer. Math. Soc. 27 (2014), no. 3, 685–705.

[13] C. Haberl and L. Parapatits, Valuations and surface area measures, J. Reine Angew. Math. 687 (2014), 225–245.

[14] H. Hadwiger, Vorlensungen über Inhalt, Oberfläche und Isoperimetrie, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1957.

[15] D. A. Klain, A short proof of Hadwiger’s characterization theorem, Mathematika 42 (1995), no. 2, 329–339.

[16] D. A. Klain, Star valuations and dual mixed volumes, Adv. Math. 121 (1996), no. 1, 80–101.

[17] D. A. Klain, Invariant valuations on star-shaped sets, Adv. Math. 125 (1997), no. 1, 95–113.

[18] D. A. Klain and G. C. Rota, Introduction to geometric probability, Cambridge University Press, Cambridge, 1997.

21 [19] B. Klartag, On convex perturbations with a bounded isotropic constant, Geom. Funct. Anal. 16 (2006), no. 6, 1271–1290.

[20] B. Klartag and E. Milman, Centroid bodies and the logarithmic Laplace transform - A uniﬁed approach, J. Funct. Anal. 262 (2012), no. 1, 10–34.

[21] H. Kone, Valuations on Orlicz spaces and Lφ-star sets, Adv. Appl. Math. 52 (2014), 82–98.

[22] J. Li, S. Yuan and G. Leng, Lp-Blaschke valuations, Trans. Amer. Math. Soc. 367 (2015), no. 5, 3161–3187.

[23] E. Lieb and M. Loss, Analysis, Graduate Studies in Mathematics, volume 14, Amer. Math. Soc., Providence, RI, 2001, 2nd edition.

[24] M. Ludwig, Projection bodies and valuations, Adv. Math. 172 (2002), no. 2, 158–168.

[25] M. Ludwig, Ellipsoids and matrix-valued valuations, Duke Math. J. 119 (2003), no. 1, 159–188.

[26] M. Ludwig, Intersection bodies and valuations, Amer. J. Math. 128 (2006), no. 6, 1409– 1428.

[27] M. Ludwig, Fisher information and matrix-valued valuations, Adv. Math. 226 (2011), no. 3, 2700–2711.

[28] M. Ludwig, Valuations on Sobolev spaces, Amer. J. Math. 134 (2012), no. 3, 827–842.

[29] M. Ludwig, Covariance matrices and valuations, Adv. Appl. Math. 51 (2013), no. 3, 359–366.

[30] M. Ludwig and M. Reitzner, A characterization of aﬀne surface area, Adv. Math. 147 (1999), no. 1, 138–172.

[31] M. Ludwig and M. Reitzner, Elementary moves on triangulations, Discrete Comput. Geom. 35 (2006), no. 4, 527–536.

[32] M. Ludwig and M. Reitzner, A classiﬁcation of SL(n) invariant valuations, Ann. Math. (2) 172 (2010), no. 2, 1219–1267.

[33] D. Ma, Real-valued valuations on Sobolev spaces, Sci. China Math. 59 (2016), no. 5, 921–934.

22 [34] P. McMullen, Valuations and dissections, in: Handbook of Convex Geometry, North- Holland, Amsterdam, 1993, volume B, 933–990.

[35] P. McMullen and R. Schneider, Valuations on convex bodies, in: Convexity and its applications, Birkh¨auser, Basel, 1983, 170–247.

q [36] M. Ober, Lp-Minkowski valuations on L -spaces, J. Math. Anal. Appl. 414 (2014), no. 1, 68–87.

[37] L. Parapatits, SL(n)-contravariant Lp-Minkowski valuations, Trans. Amer. Math. Soc. 366 (2014), no. 3, 1195–1211.

[38] L. Parapatits, SL(n)-covariant Lp-Minkowski valuations, J. London Math. Soc. (2) 89 (2014), no. 2, 397–414.

[39] R. Schneider, Convex Bodies: The Brunn-Minkowski Theory, Cambridge University Press, Cambridge, 2014, 2nd edition.

[40] R. Schneider and F. Schuster, Rotation equivariant Minkowski valuations, Int. Math. Res. Not. 72894 (2006), 1–20.

[41] F. Schuster, Valuations and Busemann-Petty type problems, Adv. Math. 219 (2008), no. 1, 344–368.

[42] F. Schuster and T. Wannerer, GL(n) contravariant Minkowski valuations, Trans. Amer. Math. Soc. 364 (2012), no. 2, 815–826.

[43] A. Tsang, Valuations on Lp-spaces, Int. Math. Res. Not. 20 (2010), 3993–4023.

[44] A. Tsang, Minkowski valuations on Lp-spaces, Trans. Amer. Math. Soc. 364 (2012), no. 12, 6159–6186.

[45] T. Wang, Semi-valuations on BV (Rn), Indiana Univ. Math. J. 63 (2014), no. 5, 1447– 1465.

[46] T. Wannerer, GL(n) equivariant Minkowski valuations, Indiana Univ. Math. J. 60 (2011), no. 5, 1655–1672.