Rotation Invariant Minkowski Classes of Convex Bodies

1. Rotation Invariant Minkowski Classes of Convex Bodies Franz Schuster Vienna University of Technology joint work with Rolf Schneider

2. Definition Notation The elements of a Minkowski class will be called -bodies. A convex body K will be called a generalized -body if there exist -bodies T1, T2 such that Let denote the space of convex bodies, i.e. compact, convex sets in n-dimensional Euclidean space , n  2. K + T1 = T2. Minkowski Classes Definition A Minkowski class is a subset of , which is closed in the Hausdorff metric and closed under Minkowski linear combinations, i.e. K, L  , ,   0  K + L  .

3. Remarks: A convex body L is an -body if and only if L can be approximated by bodies of the form 1g1K + ... + mgmK, i  0, gi  G. Invariant and Generated Minkowski Classes Definition Let G be a group of transformations of . We call a Minkowski class G-invariant if K   gK   g  G. The smallest G-invariant Minkowski class containing a given convex body K is said to be the G-invariant Minkowski class generated by K.

4. Remarks: A convex body L is an -body if and only if L can be approximated by bodies of the form A convex body L is a generalized -body if and only if h(L, . )  clspan{h(gK, . ): g  G}. 1g1K + ... + mgmK, i  0, gi  G. Invariant and Generated Minkowski Classes Definition Let G be a group of transformations of . We call a Minkowski class G-invariant if K   gK   g  G. The smallest G-invariant Minkowski class containing a given convex body K is said to be the G-invariant Minkowski class generated by K.

5. Facts: = but is nowhere dense in for n  3 2 2 c c The set of generalized zonoids is dense in for every n  2. c Zonoids and Generalized Zonoids Reminder A convex body in is called a zonoid if it can be approximated by Minkowski sums of finitely many closed line segments. Let denote the set of zonoids. A convex body K in is called a generalized zonoid if there are Z1, Z2  such that K + Z1 = Z2.

6. Fact: For n  3, the affine invariant Minkowski class generated by a convex body is nowhere dense in . c as a Minkowski Class Reminder The set of zonoids is the rigid motion invariant Minkowski class generated by a segment. is also the affine invariant Minkowski class generated by a segment. A convex body is a generalized zonoid if it is a generalized -body.

7. Theorem [Alesker, 2003]: If is the SL(n)-invariant Minkowski class generated by a non-symmetric convex body, then the set of generalized -bodies is dense in . K A3T A1T A2T SL(n)-Invariant Minkowski Classes

8. K SL(n)- vs. SO(n)-Invariant Minkowski Classes Theorem [Schneider, 1996]: LetT beatriangle.Thenthereexistsanaffinemap A such that for the SO(n)-invariant Minkowski class generated by AT, the set of generalized -bodies is dense in . AT

9. Remark: The perturbation by the linear transformation A is necessary in general, as shown by bodies of constant width (or the ball). SO(n)-Invariant Minkowski Classes Theorem [Schneider & S., 2006]: LetK benon-symmetric.Thenthereexistsalinear map A, arbitrarily close to the identity, such that for the SO(n)-invariant Minkowski class generated by AK, the set of generalized -bodies is dense in .

10. Universal Convex Bodies Definition A convex body K  is called universal, if m h(K, . )  0  m  0. Notation n Letm:C(Sn–1) denotetheorthogonal projection onto the space of spherical harmonics of dimension n and order m. m n m

11. Theorem 1 Idea of the Proof of Theorem 2: Let K be a convex body and let be the SO(n) invariant Minkowski class generated by K. Then the set of generalized -bodies is dense in if and only if K is universal. n Let {Ym1, …, YmN} be an orthonormal basis of . Then m A hAK,Ymj=hAKYmj d Sn – 1 is real analytic in a neighborhood of Id in . 2 Theorem 2 Let K  be non-symmetric. Then there exists a linear map A, arbitrarily close to the identity, such that AK is universal. Universal Convex Bodies

12. Let f  C(Sn – 1), then  ftij() d = f , YmjYmi. m SO(n) 1 N(n,m) Basic Facts on Spherical Harmonics Facts: n is invariant under the action of SO(n), thus there are real numbers tij () such that for   SO(n) m m Ymj = tij()Ymi. N(n,m) m i = 1

13. Since K is universal, cmj(m)  0 for some j(m) and every m. Define bij = if j = j(m) and 0 otherwise and let m ami g() = N(n,m)  bij tij(). k N(n,m) m m cmj(m) m = 0 i, j = 1 Result: h(K,u)g()d = h(L,u) SO(n) The Proof of Theorem 1 Definitions: Let K  be universal, cmj=hK ,Ymj and let L  , i.e. n h(L, . ) = amjYmj. k N(n,m) m = 0 j = 1