A uniqueness result for mean section bodies

Available online at www.sciencedirect.com

Advances in Mathematics 229 (2012) 596–601 www.elsevier.com/locate/aim

A uniqueness result for mean section bodies Paul Goodey a,∗ , Wolfgang Weil b a Department of Mathematics, University of Oklahoma, Norman, OK 73019, USA b Department of Mathematics, Karlsruhe Institute of Technology, 76128 Karlsruhe, Germany

Received 20 June 2011; accepted 22 September 2011 Available online 1 October 2011 Communicated by Erwin Lutwak

Abstract For 1  k  d, the k-th mean section body, Mk (K), of a convex body K in Rd , is the Minkowski sum of all its sections by k-dimensional flats. We will show that, for any fixed 1 < k < d, the body K is uniquely determined by the body Mk (K), assuming dim K  d − k + 2. This result was previously known only for centrally symmetric bodies, or, in full generality, only for the case k = 2. © 2011 Elsevier Inc. All rights reserved. Keywords: Convex bodies; Surface area measures; Mean section bodies

1. Introduction For a convex body K in Rd and k ∈ {1, . . . , d}, the mean section body Mk (K) was defined in [7] as the Minkowski average of all sections of K with k-dimensional (affine) flats. In terms of support functions, we have   h Mk (K), · =

 h(K ∩ E, ·)μk (dE),

(1.1)

A(d,k)

where A(d, k) is the affine Grassmannian and μk is the motion invariant measure on A(d, k) normalized so that the measure of all the k-flats within distance one of the origin is κd−k , the volume of the unit ball in Rd−k . * Corresponding author.

E-mail address: [email protected] (P. Goodey). 0001-8708/$ – see front matter © 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.aim.2011.09.009

P. Goodey, W. Weil / Advances in Mathematics 229 (2012) 596–601

597

In the case k = d we have Md (K) = K (κ0 = 1). The case k = 1 is also a special one. It was shown in [7] that M1 (K) is a ball whose radius is determined by the volume of K, see [4] for an alternate proof. One of the main results in [7] states that the mean section body M2 (K) determines K uniquely (up to translations), assuming dim K = d. The translational restriction was later removed in [3], where it was also shown that, for 3  k  d − 1 and centrally symmetric bodies K, the mean section body Mk (K) determines K, in case dim K  d − k + 2. In the case that k = 2 and K is origin symmetric, it follows from Corollary 2 of [7] that the support function of M2 (K) is (up to a factor) the sine transform of the surface area measure of K. This can be written in the form   ⊥   v|u Sd−1 (K, dv), h M2 (K), u = cd S d−1

since the length, v|u⊥ , of the projection of v ∈ S d−1 onto the space orthogonal to u is just the sine of the angle between u and v. Here, and in the sequel, we denote by cd and (subsequently) cd,k a positive number dependent only on the subscript(s). Its value may change from one line to the next and from one formula to another. The explicit value of this variable constant can always be calculated, but knowledge of its value does not have any significance for our discussion. Schneider [10] showed that the above integral also arises as a certain integral of surface areas of parallel hyperplane sections of K. Using Vi , for i = 0, 1, . . . , d, to denote the i-th intrinsic volume of a convex body in Rd (see [12], for example), Schneider’s result can be written in the form ∞



   Vd−2 K ∩ u⊥ + tu dt = cd

−∞

 ⊥ v|u Sd−1 (K, dv).

S d−1

This result was an extension of earlier work of Berwald [2] who established the three-dimensional version. Further applications of the sine transform can be found in the work of Maresch and Schuster [9]. It was shown in [7] that, if K is centrally symmetric, the support function of Mk (K) d is (up to normalization) the Radon transform Rd+1−k,1 of the (d + 1 − k)-th projection function of K. As observed in [3], this can be combined with results of Schneider [11], Schneider and Weil [14] and Goodey, Schneider and Weil [6] to see that, for centrally symmetric K,   h Mk (K), u = cd,k

∞

   Vd−k K ∩ u⊥ + tu dt.

−∞

For arbitrary bodies K, the mean section bodies arise naturally in integral geometry. We mention just one instance, see [15, Theorem 6.4.7] for example. The centred support function h∗ (K, ·) denotes the support function of K after the body is translated to have its Steiner point at the origin o. Then, if Gd denotes the group of rigid motions in Rd , we have  Gd

h∗ (K ∩ gM, ·)μ(dg) =

d  k=1

  h∗ Mk (K), · Vk (M),

598

P. Goodey, W. Weil / Advances in Mathematics 229 (2012) 596–601

for arbitrary convex bodies K and M, where μ is the Haar measure on Gd normalized so that, for all x ∈ Rd and all convex bodies K ⊂ Rd , μ{g: gx ∈ K} = Vd (K), the volume of K. In general, it remained unknown whether Mk (K), for 3  k  d − 1, uniquely determines the body K. It is the main purpose of this work to settle this uniqueness problem. Our approach here will be to use the spherical lifting and projection operators, introduced in [5], to reduce the case of k-dimensional sections to the case k = 2, for which uniqueness is known. In Section 2, we collect some preliminary material including an integral formula which shows the interplay between our mean section operation Mk and the (d +1−k)-th surface area measures of the bodies concerned. This will be the key to the reduction argument in Section 3. We refer the reader to [12] for all general notions from convex geometry which are used in the following. This includes, in particular, mixed volumes, intrinsic volumes and surface area measures. 2. Preliminaries Let K denote the set of all convex bodies in Rd . For a polytope K ∈ K and k ∈ {1, . . . , d − 1}, it was shown in [7] that   h∗ Mk (K), u = cd,k



  γ (F, K; −u) F, u Vd+1−k (F ),

(2.1)

F ∈Fd+1−k (K)

for u in the unit sphere S d−1 . Here, Fd+1−k (K) is the set of (d + 1 − k)-dimensional faces of K, γ (F, K; −u) is a common outer angle, | F, u | is the length of the projection of u onto the linear space parallel to the affine hull of F . Formula (2.1) is also true for k = d and was proved in this case in [16] (see also [13], for a shorter proof). It follows from (2.1) that the mapping   Hk : K → h∗ Mk (K), · on K is homogeneous of degree d + 1 − k. For polytopes K this is immediate, for general bodies it follows by approximation. The mapping Hk is also additive, hence it gives rise to a Minkowski valuation ϕk : K → Mk∗ (K), where Mk∗ (K) is the centred version of Mk (K) defined by h(Mk∗ (K), ·) = h∗ (Mk (K), ·). In fact, from (1.1) we obtain         h∗ Mk (K ∪ L), · + h∗ Mk (K ∩ L), · = h∗ Mk (K), · + h∗ Mk (L), · , for K, L, K ∪ L ∈ K, since K → h(K, u) is a (real-valued) valuation, for each u ∈ Rd . The following theorem will be crucial for the uniqueness result which we will prove in the next section. It is a symmetry relation between the support function h(Mk (K), ·) of the mean section body Mk (K) and the (d + 1 − k)-th surface area measure Sd+1−k (L, ·) of a body L ∈ K. We obtain it as a consequence of work of Alesker, Bernig and Schuster [1] on bivaluations. Theorem 2.1. For convex bodies K, L and k ∈ {2, . . . , d}, we have       h Mk (K), u Sd+1−k (L, du) = h Mk (L), u Sd+1−k (K, du). S d−1

S d−1

P. Goodey, W. Weil / Advances in Mathematics 229 (2012) 596–601

599

Proof. We first remark that the assertion can be reformulated in terms of mixed volumes as     V Mk∗ (K)[1], L[d + 1 − k], B d [k − 2] = V Mk∗ (L)[1], K[d + 1 − k], B d [k − 2] ,

(2.2)

where B d denotes the unit ball and where we have used standard notations and properties of mixed volumes from [12]. We also replaced Mk (K), Mk (L) by the centred versions Mk∗ (K), Mk∗ (L). As we already mentioned, ϕk is a Minkowski valuation, homogeneous of degree d + 1 − k. Since   ∗  h Mk (K), · = h∗ (K ∩ E, ·)μk (dE), A(d,k)

we obtain, in addition, that ϕk is translation invariant, continuous and covariant with respect to rotations ϑ ∈ SO(d). The symmetry relation (2.2) is thus a direct consequence of Corollary 7.2 in [1]. 2 3. The uniqueness theorem Theorem 3.1. For 2  k  d, a convex body K ⊂ Rd of dimension dim K  d + 2 − k is uniquely determined by the mean section body Mk (K). Proof. We first show determination up to translations. The case k = 2 was settled in [7], hence we now assume k > 2. Then, 2  d + 2 − k  d − 1. We consider a (d + 2 − k)-dimensional convex body L. From Theorem 2.1, we obtain 

  h Mk (K), u Sd+1−k (L, du) =

S d−1



  h Mk (L), u Sd+1−k (K, du).

S d−1

Let H be the affine hull of L (we may assume o ∈ H ). Then, almost all affine k-flats E in Rd intersect H in a two-dimensional plane and the image of the invariant measure μk on A(d, k) under E → E ∩ H is an invariant measure μH 2 on the space A(H, 2) of affine planes in H . Therefore, we have   h Mk (L), u =



A(d,k)

 h(L ∩ E, u)μk (dE) =

 ∗  H  = cd,k πH,1 h M2 (L), · (u),

    h L ∩ E  , u μH 2 dE

A(H,2)

∗ where M2H (L) is the mean section body of L of order 2 in H and πH,1 is a lifting operator ∗ considered in [5] (and in [4] with a different notation). The operator πH,1 lifts support functions of convex bodies in H to support functions of the same bodies in Rd . So, it follows from formula (5.4) in [5] that

600

P. Goodey, W. Weil / Advances in Mathematics 229 (2012) 596–601



  h Mk (L), u Sd+1−k (K, du) = cd,k

S d−1



 ∗  H  πH,1 h M2 (L), · (u)Sd+1−k (K, du)

S d−1



= cd,k

  h M2H (L), u πH,1 Sd+1−k (K, ·) (du),

S d−1 ∩H

where πH,1 is the adjoint spherical projection operator. For given K, the knowledge of Mk (K) implies the knowledge of 

  h Mk (K), u Sd+1−k (L, du)

S d−1

for all convex bodies L, in particular for all bodies L of dimension d + 2 − k. Therefore, for every subspace H of dimension d + 2 − k, we know all integrals 

  h M2H (L), u πH,1 Sd+1−k (K, ·) (du)

(3.1)

S d−1 ∩H

for convex bodies L ⊂ H . Since the measure πH,1 Sd+1−k (K, ·) annihilates linear functions, we may replace here h(M2H (L), ·) by h∗ (M2H (L), ·). Using a Hahn–Banach argument, it is explained in [8] (see the comments preceding Theorem 5.5), that, because of the uniqueness in case k = 2, ∞ d−1 ∩ H ) → C ∞ (S d−1 ∩ H ), which can be extended there is a continuous bijection mH o 2 : Co (S H ∗ ∗ to centred support functions of convex bodies in H , and satisfies mH 2 h (L, ·) = h (M2 (L), ·) ∞ d−1 ∩ H ) denotes the space of smooth functions f on for all convex bodies L ⊂ H ; here Co (S S d−1 ∩ H with centroid at the origin,  xf (x) dx = o. S d−1 ∩H

It follows, therefore, from (3.1) that we know the integrals 

 f (u) πH,1 Sd+1−k (K, ·) (du)

  for all f ∈ Co∞ S d−1 ∩ H .

S d−1 ∩H

Consequently, we know the measure πH,1 Sd+1−k (K, ·) for every H ∈ G(d, d + 2 − k). It was proved in Corollary 3.4 of [4] that Sd+1−k (K, ·) is, therefore, uniquely determined (note that the projection operator πH considered there agrees with our operator πH,1 ). Finally, for a body K with dim K  d +2−k, the surface area measure Sd+1−k (K, ·) determines K (up to translations). We have proved that, if Mk (K1 ) = Mk (K2 ) then there is a translation t ∈ Rd such that K2 = K1 + t, assuming K1 , K2 have dimension at least d − k + 2. It is shown in Section 5 of [3] that we would then have Mk (K1 ) = Mk (K1 + t) = Mk (K1 ) + cd,k Vd−k (K1 )t.

P. Goodey, W. Weil / Advances in Mathematics 229 (2012) 596–601

601

The dimensional restriction on the bodies implies that Vd−k (K1 ) > 0 and so we deduce that t = o as required. 2 References [1] S. Alesker, A. Bernig, F. Schuster, Harmonic analysis of translation invariant valuations, Geom. Funct. Anal. 21 (2011) 751–773. [2] L. Berwald, Integralgeometrie 25. Über die Körper konstanter Helligkeit, Math. Z. 42 (1937) 737–738. [3] P. Goodey, Radon transforms of projection functions, Math. Proc. Cambridge Philos. Soc. 123 (1998) 159–168. [4] P. Goodey, M. Kiderlen, W. Weil, Section and projection means of convex bodies, Monatsh. Math. 126 (1998) 37–54. [5] P. Goodey, M. Kiderlen, W. Weil, Spherical projections and liftings in geometric tomography, Adv. Geom. 11 (2011) 1–47. [6] P. Goodey, R. Schneider, W. Weil, Projection Functions on Higher Rank Grassmannians, Oper. Theory Adv. Appl., vol. 77, Birkhäuser, 1995, pp. 75–90. [7] P. Goodey, W. Weil, The determination of convex bodies from the mean of random sections, Math. Proc. Cambridge Philos. Soc. 112 (1992) 419–430. [8] P. Goodey, W. Weil, Local properties of intertwining operators on the sphere, Adv. Math. 227 (2011) 1144–1164. [9] G. Maresch, F. Schuster, The sine transform of isotropic measures, Int. Math. Res. Notices (2011), doi: 10.1093/imrn/rnr035, in press. [10] R. Schneider, Über eine Integralgleichung in der Theorie der konvexen Körper, Math. Nachr. 44 (1970) 55–75. [11] R. Schneider, Crofton’s formula generalized to projected thick sections, Rend. Circ. Mat. Palermo 30 (1981) 157– 160. [12] R. Schneider, Convex Bodies: The Brunn–Minkowski Theory, Cambridge University Press, Cambridge, 1993. [13] R. Schneider, Mixed polytopes, Discrete Comput. Geom. 29 (2003) 575–593. [14] R. Schneider, W. Weil, Translative and kinematic integral formulae for curvature measures, Math. Nachr. 129 (1986) 67–80. [15] R. Schneider, W. Weil, Stochastic and Integral Geometry, Springer, Berlin, 2008. [16] W. Weil, Translative and kinematic integral formulae for support functions, Geom. Dedicata 57 (1995) 91–103.