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

2. Minkowski Classes Def inition
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 
Def inition
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
K + T1 = T2.
Notation
Let denote the space of convex bodies, i.e. compact, convex sets in n-dimensional Euclidean space , n  2.

3. Invariant and Generated Minkowski Classes
Def inition
Let G be a group of transformations of We call a Minkowski class G-invariant if
K   g K   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.
Remarks:
A convex body L is an body if and only if L can be approximated by bodies of the form
1 g1 K m gm K, i  0, gi  G.

4. h(L, . )  cl span {h( g K, . ): g  G}.
Invariant and Generated Minkowski Classes
Def inition
Let G be a group of transformations of We call a Minkowski class G-invariant if
K   g K   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.
Remarks:
A convex body L is a generalized body if and only if
h(L, . )  cl span {h( g K, . ): g  G}.
A convex body L is an body if and only if L can be approximated by bodies of the form
1 g1 K m gm K, i  0, gi  G.

5. 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.
Facts:
= but is nowhere dense in for n  3 2 c
The set of generalized zonoids is dense in for every n  2. c

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

7. K SL(n)-Invariant Minkowski Classes 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
A1T
A2T
A3T

8. K SL(n)- vs. SO(n)-Invariant Minkowski Classes
Theorem [Schneider, 1996]:
Let T  be a triangle. Then there exists an affine map A such that for the SO(n)-invariant Minkowski class
generated by AT, the set of generalized -bodies is dense in K AT

9. SO(n)-Invariant Minkowski Classes
Theorem [Schneider & S., 2006]:
Let K  be non-symmetric. Then there exists a linear 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 .
Remark:
The perturbation by the linear transformation A is necessary in general, as shown by bodies of constant width (or the ball).

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

11. Universal Convex Bodies
Theorem 1
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.
Idea of the Proof of Theorem 2:
Let {Ym1, …, YmN} be an orthonormal basis of
Then
is real analytic in a neighborhood of Id in m n
A  hAK , Ymj =  hAK Ymj d 2
S n – 1
Theorem 2
Let K  be non-symmetric. Then there exists a linear map A, arbitrarily close to the identity, such that AK is universal.

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

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