Structural Theory of Addition and Symmetrization in Convex Geometry

1. Structural Theory of Addition and Symmetrization in Convex Geometry Richard Gardner www.geometrictomography.com

2. Some Properties Commutativity: Associativity: Homogeneity of degree k: Monotonicity: Identity: Continuity: GL(n) covariance: Projection covariance: Note that for compact convex sets, continuity and GL(n) covariance implies projection covariance.

3. Theorem 1 [GHW 2] is projection covariant iff where M is a 1-unconditional compact convex set in An Orlicz-Brunn-Minkowski theory was initiated by LYZ (also Ludwig, Reitzner) in 2010. E. Lutwak, D. Yang, and G. Zhang, Orlicz projection bodies,Adv. Math. 223 (2010), 220-242. E. Lutwak, D. Yang, and G. Zhang, Orlicz centroid bodies,J. Differential Geom. 84 (2010), 365-387. C. Haberl, E. Lutwak, D. Yang, and G. Zhang, The even Orlicz Minkowski problem,Adv. Math. 224 (2010), 2485-2510. R.J.G., D. Hug,and W. Weil, The Orlicz-Brunn-Minkowski theory: A general framework, additions, and inequalities,J. Differential Geom. 97 (2014), 427-476. [GHW2]

4. Orlicz Addition [GHW2, XJL] Consider the set Φmof convex functions φ: [0,∞)m → [0,∞), increasing in each variable, such that φ(o) = 0 and φ(ej)=1 for j = 1,…, m. If m ≥ 2, define by The operation +φis well defined, monotonic, continuous, GL(n) covariant, (hence projection covariant) and Normalization, identity property!

5. Why Not a Simpler Definition? Recall Lp addition: ? NO! [Theorem 10.1, GHW2] implies that φ(t) = tp for p ≥ 1. This uses associativity and earlier results. In fact [Theorem 5.10, GHW2] states that Orlicz addition is associative if and only if it is Lp addition for 1 ≤ p ≤ ∞. [Theorem 5.9, GHW2] gives necessary and sufficient conditions on φfor +φ to be commutative.

6. General Framework [GHW2] For with define the Orlicz norm of f with respect to µ. Take and for , let Then defines a compact convex set C(φ,µ) in

7. Applications [GHW2] If we get Orlicz addition. Defining, for each K, µ = γK by Defining, for each K, µ = πK by yields C(φ, µ) = Γ φK, the Orlicz centroid body. yields C(φ, µ) = Π φK, the Orlicz projection body. D. Xi, H. Jin,and G. Leng, The Orlicz-Brunn-Minkowski inequality,Adv. Math. 260 (2014), 350-374. [XJL]

8. Applications [GHW2] If we get Orlicz addition. Project: What other choices of μ lead to interesting and useful sets C(φ, µ) ?

9. Relation to M-Addition For let Jφ be the 1-unconditional convex body in defined by By [Theorem 5.3, GHW2], is M-addition with and is M-addition with Moreover, IfM is 1-unconditional with e1,…, em in its boundary, then the converse holds with

10. Extension to Arbitrary Sets One might define by or equivalently by A different extension, akin to that of LYZ for Lp addition, is considered in [Section 6, GHW2], but only when m = 2. Project: Investigate fully the possibility of extending Orlicz addition so that Is there any relationship with M-addition?

11. Extensions of the BM Theory So what is next? K. Böröczky, E. Lutwak, D. Yang, and G. Zhang, The log-Brunn-Minkowski inequality,Adv. Math. 231 (2012), 1974-1997.

12. Radial Addition Radial function E. Lutwak, Dual mixed volumes, Pacific J. Math. 58 (1975), 531-538.

13. pth radial addition: Firey (1961), Falconer (1983), R.J.G. (1987); see [Chapter 6, G]. radial Orlicz addition: Additions in the Dual BM Theory B. Zhu, J. Zhou,and W. Xu, Dual Orlicz-Brunn-Minkowski theory,Adv. Math. 264 (2014), 700-725. R.J.G., D. Hug, W. Weil, and D. Ye, The dual Orlicz-Brunn-Minkowski theory,J. Math. Anal. Appl. 430 (2015), 810-829. (also radial M-addition.)

14. i-Symmetrizations Let An i-symmetrization on a class of compact sets in is a map where is the class of sets in that are H-symmetric, i.e. symmetric with respect to the (fixed) i-dimensional subspace H. Monotonicity: Projection invariance: Invariance on H-symmetric sets: Invariance on H-symmetric spherical cylinders:

15. Central Symmetrization Central symmetral of K Characterizations in [Section 8, GHW1] and in J. Abardia-Evéquoz and E. Saorín Gómez, The role of the Rogers-Shephard inequality in the characterization of the difference body, Forum Math. 29 (2017), 1227–1243. .

16. Steiner Symmetrization Let H be an (n-1)-dimensional subspace in and let be an (n-1)-symmetrization. If is monotonic, volume preserving, and eitherinvariant on H-symmetric spherical cylinders orprojection invariant, then for each K not contained in a hyperplane orthogonal to H. Various examples show that none of the assumptions in these results can be omitted. G. Bianchi, R.J.G.,and P. Gronchi, Symmetrization in geometry,Adv. Math. 306 (2017), 51-88. [BGG1]

17. MinkowskiSymmetrization Minkowski symmetral of K: (If i = 0, then )

18. MinkowskiSymmetrization 1 Let H be an (n-1)-dimensional subspace in and let be an (n-1)-symmetrization. If is monotonic, mean width preserving, and eitherinvariant on H-symmetric spherical cylinders orprojection invariant, then is Minkowski symmetrization with respect to H. Various examples show that none of the assumptions in these results can be omitted.

19. MinkowskiSymmetrization 2 Let let H be an i-dimensional subspace in and let be an i-symmetrization. If is monotonic, invariant on H-symmetric sets, andinvariant under translations orthogonal to H of H-symmetric sets, then and if i = 0, then If in addition is mean width preserving, then is Minkowski symmetrization with respect to H. ?

20. Convergence Let and let be a symmetrization process on . Suppose that for each i-dimensional subspace H, is monotonic, invariant on H-symmetric sets, and invariant under translations orthogonal to H of H-symmetric sets. If (Hm) is an M-universal sequence, then (Hm)is weakly -universal. ? Example: monotonic + invariant on H-symmetric sets + M-universal -universal. G. Bianchi, R.J.G.,and P. Gronchi, Convergence of symmetrization processes,in preparation. [BGG2]

21. Klain’s Theorem [BGG2] Let (Hm) be a sequence of (n-1)-dimensional subspaces chosen from a finite set , each appearing infinitely often. Then for every , the successive Steiner symmetrals converge to a compact convex set that is symmetric with respect to each subspace in . Klain’s theorem holds when , where H is i-dimensional, , is monotonic,invariant on H-symmetric sets, invariant under translations orthogonal to H of H-symmetric sets, and continuous. D. Klain, Steiner symmetrization using a finite set of directions,Adv. in Appl. Math. 48 (2012), 340-353.

22. Corollaries [BGG2] 1. For each , Klain’s theorem holds for fiber and Minkowski symmetrization. 2. For each , it also holds for Schwarz symmetrization, in which case the limiting convex body is rotationally symmetric with respect to each subspace in . 3. Alternating Schwarz symmetrizations with respect to two lines through o in converge to a ball. G. Bianchi, A. Burchard, P. Gronchi, and A. Volčič, Convergence in shape of Steiner symmetrization,Indiana Univ. Math. J. 61 (2012), 1695-1710. L. Tonelli, Sulla proprietá di minimo della sfera,Rend. Cir. Math. Palermo 39 (1915), 109-138.

23. Open Problem [BGG1] Let and let H be an (n-1)-dimensional subspace in . Is there an (n-1)-symmetrization that is monotonic, Vj-preserving, and eitherinvariant on H-symmetric spherical cylinders orprojection invariant? No for a modified version if What about j = n-1? C. Saroglou, On some problems concerning symmetrization operators,preprint.

24. End of Part 2Note: The talk in Castro Urdiales 2018 ended here. The following few slides supplement a few topics touched on in the talk.

25. Projection Bodies Petty Projection inequality: nV(K) with equality iff K is an ellipsoid.

26. Orlicz Projection Bodies Convex functions φ: → [0,∞) such that φ(0) = 0. for , where dVK(u) = hK(u) dS(K,u). Orlicz Petty Projection inequality: E. Lutwak, D. Yang, and G. Zhang, Orlicz projection bodies,Adv. Math. 223 (2010), 220-242.

27. Centroid Bodies Busemann-Petty Centroid inequality: with equality iff K is an o-symmetric ellipsoid.

28. Orlicz Centroid Bodies for (star bodies with continuous and positive radial functions). Orlicz Busemann-Petty Centroid inequality: for . E. Lutwak, D. Yang, and G. Zhang, Orlicz centroid bodies,J. Differential Geom. 84 (2010), 365-387.

29. The Dual Brunn-Minkowski Theory Combine radial addition and volumeV. Lutwak’s Theorem on Dual Mixed Volumes: IfKi, i = 1,…, m, are star sets in , ti ≥ 0,i = 1,…, m, and then

30. Lutwak’s Dictionary

31. Steiner Symmetrization Steiner symmetral SHK of K Also for compact sets…

32. Properties and Applications • Preserves volume. • Generally reduces surface area. • There is a sequence of directions in Sn-1 such that the corresponding successive Steiner symmetrals of a convex body K in converge to a ball with center at the origin. • This (with Blaschke’s Selection Theorem) yields the isoperimetric inequality for convex bodies in : • Manyapplications to inequalities, PDEs, math. physics. B. Kawohl, Rearrangement of Level Sets in PDE, Springer, Berlin, 1985. G. Pólya and G. Szegö, Isoperimetric Inequalities in Mathematical Physics, Princeton Univ. Press, Princeton, NJ, 1951.

33. Other Symmetrizations Schwarz symmetral SHK of K (replace (n-i)-dimensional sections by (n-i)-dimensional balls of the same (n-i)- dimensional volume). Blaschke-Minkowski symmetral of K: Note that and Also Lp and M-addition versions, Blaschke symmetrization, …

34. Fiber Symmetrization Steiner central P. McMullen, New combinations of convex sets,Geom. Dedicata 78 (1999), 1-19.

35. Properties of Known Symmetrizations