1. Franz SchusterValuations and Busemann-Petty Type Problems Valuations and Busemann-Petty Type Problems Franz Schuster Vienna University of Technology

2. Franz SchusterValuations and Busemann-Petty Type Problems The Busemann-Petty Problem Notation Let denote the space of o-symmetric convex bodies in, n  3. H. Busemann & C. Petty 1956 Let K, L . e e for all u  S n – 1. Does it follow that V(K )  V(L)? V n – 1 (K  u  )  V n – 1 (L  u  ), u u  KL u u  Suppose that

3. Franz SchusterValuations and Busemann-Petty Type Problems The Busemann-Petty Problem H. Busemann & C. Petty 1956 Let K, L . Suppose that for all u  S n – 1. Does it follow that V(K )  V(L)? V n – 1 (K  u  )  V n – 1 (L  u  ), u u  KL e u u  G. C. Shephard 1964 Let K, L . Suppose that for all u  S n – 1. Does it follow that V(K )  V(L)? V n – 1 (K | u  )  V n – 1 (L | u  ), e u u  K u u  L and the Shephard Problem

4. Franz SchusterValuations and Busemann-Petty Type Problems The Busemann-Petty Problem H. Busemann & C. Petty 1956 Let K, L . Suppose that for all u  S n – 1. Does it follow that V(K )  V(L)? V n – 1 (K  u  )  V n – 1 (L  u  ), u u  KL e u u  G. C. Shephard 1964 Let K, L . Suppose that for all u  S n – 1. Does it follow that V(K )  V(L)? V n – 1 (K | u  )  V n – 1 (L | u  ), e and the Shephard Problem Petty, Schneider 1965 The solution to Shephard's problem is negative if n  3.

5. Franz SchusterValuations and Busemann-Petty Type Problems The Solution of the Busemann-Petty Problem Contributions by … K. Ball 1986: Negative answer for n  10 A. Giannopolous 1990, J. Bourgain 1991: Negative answer for n  7 M. Papadimitrakis 1992: Negative answer for n  5 D. Larman und C. A. Rogers 1975: Negative answer for n  12 E. Lutwak 1989: Duality Shephard- and Busemann-Petty problem G. Zhang 1999: Affirmative answer for n = 4 R. Gardner 1994: Affirmative answer for n = 3 R. Gardner, A. Koldobsky, T. Schlumprecht 1999: New Proof

6. Franz SchusterValuations and Busemann-Petty Type Problems The Busemann-Petty Problem H. Busemann & C. Petty 1956 Let K, L . Suppose that for all u  S n – 1. Does it follow that V(K )  V(L)? V n – 1 (K  u  )  V n – 1 (L  u  ), e G. C. Shephard 1964 Let K, L . Suppose that for all u  S n – 1. Does it follow that V(K )  V(L)? V n – 1 (K | u  )  V n – 1 (L | u  ), e and the Shephard Problem Petty, Schneider 1965 The solution to Shephard's problem is negative if n  3. Many mathematicians - 1999 The solution to the Busemann Petty problem is affirmative if n = 3, 4 and negative if n  5.

7. Franz SchusterValuations and Busemann-Petty Type Problems h(  K,u) = V n – 1 (K |u  ) The projection body operator  :  Projection Bodies h(K,u) = max{u. x: x  K} Support functions o - h(K,- u) u K h(K,u)h(K,u) K  L  h(K,. )  h(L,. ) A simple fact:

8. Franz SchusterValuations and Busemann-Petty Type Problems The Busemann-Petty Problem H. Busemann & C. Petty 1956 Let K, L . Suppose that for all u  S n – 1. Does it follow that V(K )  V(L)? V n – 1 (K  u  )  V n – 1 (L  u  ), e G. C. Shephard 1964 Let K, L . Suppose that V(K )  V(L)? for all u  S n – 1. Does it follow that V n – 1 (K | u  )  V n – 1 (L | u  ), e and the Shephard Problem Petty, Schneider 1965 The solution to Shephard's problem is negative if n  3. Many mathematicians - 1999 The solution to the Busemann Petty problem is affirmative if n = 3, 4 and negative if n  5.  K   L. K   L. Does it follow that

9. Franz SchusterValuations and Busemann-Petty Type Problems Intersection Bodies  (IL,u) = V n – 1 (L  u  ) Def inition of I:  o L (L,u)(L,u) Radial functions K  L   (K,. )   (L,. ) A simple fact:  (L,u) = max{ ≥ 0: u  L}

10. Franz SchusterValuations and Busemann-Petty Type Problems The Busemann-Petty Problem H. Busemann & C. Petty 1956 Let K, L . Suppose that V(K )  V(L)? for all u  S n – 1. Does it follow that V n – 1 (K  u  )  V n – 1 (L  u  ), e G. C. Shephard 1964 Let K, L . Suppose that V(K )  V(L)? e and the Shephard Problem Petty, Schneider 1965 The solution to Shephard's problem is negative if n  3. Many mathematicians - 1999 The solution to the Busemann Petty problem is affirmative if n = 3, 4 and negative if n  5.  K   L. K   L. Does it follow that I K  I L.I K  I L.

11. Franz SchusterValuations and Busemann-Petty Type Problems Variants of the Busemann-Petty Problem Contributions by … E. Lutwak: Centroid body operator  Negative answer for n  3 D. Ryabogin, A. Zvavitch: L p -Projection body operator  p Negative answer for n  2, p > 1 V. Yaskin, M. Yaskina: L p -Polar Centroid body operator  p * Negative answer for n  3, 0 < p < 1 Affirmative answer for n = 3, – 1 < p  0, Negative answer for n  4 C. Haberl: L p -Intersection body operator I p +

12. Franz SchusterValuations and Busemann-Petty Type Problems Valuations Def inition A map  :  A into an abelian semigroup (A,+) is called a valuation if  (K  L) +  (K  L) =  (K  (L  (K  L) +  (K  L) =  (K ) +  (L) whenever K, L and K  L . Co- or Contravariant Valuations A valuation  on taking subsets of as values is called GL(n) if there exists a q , such that  (AK ) = | det A | q A  K,  K , A  GL(n). covariant

13. Franz SchusterValuations and Busemann-Petty Type Problems Valuations Def inition A map  :  A into an abelian semigroup (A,+) is called a valuation if  (K  L) +  (K  L) =  (K  (L  (K  L) +  (K  L) =  (K ) +  (L) whenever K, L and K  L . Co- or Contravariant Valuations A valuation  on taking subsets of as values is called GL(n) if there exists a q , such that  (AK ) = | det A | q A  K,  K , A  GL(n). contravariant covariant - T- T

14. Franz SchusterValuations and Busemann-Petty Type Problems Theorem [Ludwig, 2005] A map  :  is a continuous Minkowski valuation which is GL(n) contravariant if and only if  = c  for some c  0. Theorem [Ludwig, 2006] A map  :  is a continuous valuation with respect to radial addition which is GL(n) contravariant if and only if  = c I for some c  0. Classif ication of Projection and Intersection Bodies

15. Franz SchusterValuations and Busemann-Petty Type Problems The Question Let  be a continuous valuation defined on subsets of with values in the space of convex or star bodies. Suppose that Valuations and Busemann-Petty Type Problems does it follow that V(K )  V(L)?  L, L, L, L,  K K K K  Example Identity, reflection in the origin K V(K )B

16. Franz SchusterValuations and Busemann-Petty Type Problems SO(n) equivariant and translation invariant Let  :  be a continuous Minkowski valuation which is (n – 1)-homogeneous In the following … A Variant of Shephard's Problem Example Projection body operator 

17. Franz SchusterValuations and Busemann-Petty Type Problems First observation If  is not injective, the answer is no, in general. A Variant of Shephard's Problem The Question Let  be a continuous Minkowski valuation, SO(n) equivariant, translation invariant, (n – 1)-homogeneous. does it follow that V(K )  V(L)?  L, L,  K K  Notation Let (  ) denote the injectivity set of . Let K, L  (  ) and suppose that Suppose that

18. Franz SchusterValuations and Busemann-Petty Type Problems Theorem [S. 2006] Schneider-Lutwak Connection Let  be a continuous Minkowski valuation, SO(n) equivariant, translation invariant, (n – 1)-homogeneous. Let  (  ). Then  L, L,  K K  for K, L  always implies V(K )  V(L) if and only if every M  with h(M,. )  is contained in . e e e e Theorem [S. 2005] The set  is nowhere dense in. Basic Tool [S. 2005] h(  K,. ) = S(K,. ) * h(F,. )

19. Franz SchusterValuations and Busemann-Petty Type Problems SO(n) equivariant Let  :  be a continuous map which is In the following … A Variant of the Busemann-Petty Problem Radial Blaschke-Minkowski additive, i.e.  (K L) =  K  L n - 1 Let  :  be a continuous valuation, homogeneous of degree n – 1. Then Theorem [Klain 1996]  (K L) =  (K ) +  (L). n - 1 Example Intersection body operator I

20. Franz SchusterValuations and Busemann-Petty Type Problems First observation If  is not injective, the answer is no, in general. The Question Let  :  be a continuous map, SO(n) equivariant and radial Blaschke-Minkowski additive. does it follow that V(K )  V(L)?  L, L,  K K  Let K, L  (  ) and suppose that A Variant of the Busemann-Petty Problem

21. Franz SchusterValuations and Busemann-Petty Type Problems Theorem [S. 2006] Schneider-Lutwak Connection  L, L,  K K  for K, L  always implies Let  :  be a continuous map, SO(n) equivariant and radial Blaschke-Minkowski additive. Let  (  ). Then V(K )  V(L) if and only if every M  with  (M,. )  is contained in . e e e e Problem Determine the "size" of .

22. Franz SchusterValuations and Busemann-Petty Type Problems