1. Structural Theory of Addition and Symmetrization in Convex Geometry
Richard Gardner
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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.
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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),
E. Lutwak, D. Yang, and G. Zhang, Orlicz centroid bodies, J. Differential Geom. 84 (2010),
C. Haberl, E. Lutwak, D. Yang, and G. Zhang, The even Orlicz Minkowski problem, Adv. Math. 224 (2010),
R.J.G., D. Hug, and W. Weil, The Orlicz-Brunn-Minkowski theory: A general framework, additions, and inequalities, J. Differential Geom. 97 (2014),
[GHW2]

4. Orlicz Addition [GHW2, XJL]
Consider the set Φm of 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 +φ is well defined, monotonic, continuous, GL(n) covariant, (hence projection covariant) and
Normalization, identity property!
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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 φ for +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 = πK by
Defining, for each K, µ = γK by
yields C(φ, µ) = Π φK, the Orlicz projection body.
yields C(φ, µ) = Γ φK, the Orlicz centroid body.
D. Xi, H. Jin, and G. Leng, The Orlicz-Brunn-Minkowski inequality, Adv. Math. 260 (2014),
[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,
If M is 1-unconditional with e1,…, em in its boundary, then the converse holds with
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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?
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11. Extensions of the BM Theory
Type of theory
Properties lost by addition between o-symmetric compact convex sets
Classical BM theory
Lp BM theory, p > 1
Algebraic: distributivity Geometric: translation invariance
Orlicz BM theory
Algebraic: commutativity, associativity
Further extensions
Geometric: Either continuity or GL(n) covariance
So what is next?
K. Böröczky, E. Lutwak, D. Yang, and G. Zhang, The log-Brunn-Minkowski inequality, Adv. Math. 231 (2012),
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12. Radial Addition Radial function
E. Lutwak, Dual mixed volumes, Pacific J. Math. 58 (1975),
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13. Additions in the Dual BM Theory
pth radial addition: Firey (1961), Falconer (1983), R.J.G. (1987); see [Chapter 6, G].
radial Orlicz addition:
B. Zhu, J. Zhou, and W. Xu, Dual Orlicz-Brunn-Minkowski theory, Adv. Math. 264 (2014),
R.J.G., D. Hug, W. Weil, and D. Ye, The dual Orlicz-Brunn-Minkowski theory, J. Math. Anal. Appl. 430 (2015),
(also radial M-addition.)
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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 either invariant on H-symmetric spherical cylinders or projection 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),
[BGG1]
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17. Minkowski Symmetrization
Minkowski symmetral of K:
(If i = 0, then )
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18. Minkowski Symmetrization 1
Let H be an (n-1)-dimensional subspace in and let be an (n-1)-symmetrization. If is monotonic, mean width preserving, and either invariant on H-symmetric spherical cylinders or projection invariant, then is Minkowski symmetrization with respect to H.
Various examples show that none of the assumptions in these results can be omitted.
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19. Minkowski Symmetrization 2
Let let H be an i-dimensional subspace in and let be an i-symmetrization. If is monotonic, invariant on H-symmetric sets, and invariant 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. ?
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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]
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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),

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),
L. Tonelli, Sulla proprietá di minimo della sfera, Rend. Cir. Math. Palermo 39 (1915),

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 either invariant on H-symmetric spherical cylinders or projection invariant?
No for a modified version if What about j = n-1?
C. Saroglou, On some problems concerning symmetrization operators, preprint.
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24. End of Part 2 Note: The talk in Castro Urdiales 2018 ended here
End of Part 2 Note: The talk in Castro Urdiales 2018 ended here. The following few slides supplement a few topics touched on in the talk.
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25. Projection Bodies Petty Projection inequality: nV(K)
with equality iff K is an ellipsoid.
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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),

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

29. The Dual Brunn-Minkowski Theory
Combine radial addition and volume V.
Lutwak’s Theorem on Dual Mixed Volumes:
If Ki, i = 1,…, m, are star sets in ,
ti ≥ 0, i = 1,…, m, and
then
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30. Lutwak’s Dictionary Convex bodies Star bodies Projections
Sections through o
Support function hK
Radial function ρK
Brightness function bK
Section function sK
Projection body ΠK
Intersection body IK
Cosine transform
Spherical Radon transform
Surface area measure
ρKn-1
Mixed volumes
Dual mixed volumes
Brunn-Minkowski ineq.
Dual B-M inequality
Petty Projection ineq.
Busemann Intersection ineq.
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31. Steiner Symmetrization
Steiner symmetral SHK of K
Also for compact sets…
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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 :
Many applications 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.
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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.
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35. Properties of Known Symmetrizations
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