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Structural Theory of Addition and Symmetrization in Convex Geometry

This article explores the theory of addition and symmetrization in convex geometry, specifically focusing on Minkowski addition and the Brunn-Minkowski theory. It discusses the properties and applications of these binary operations and their relationship to Lp addition. The article also introduces M-addition and M-combination and highlights their significance in convex geometry.

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Structural Theory of Addition and Symmetrization in Convex Geometry

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

  2. Minkowski Addition

  3. The Brunn-Minkowski Theory Combine Minkowski addition + and volumeV. Minkowski’s Theorem on Mixed Volumes: IfKi, i = 1,…, m, are compact convex sets in , ti ≥ 0,i = 1,…, m, and then

  4. Lp Addition For 1< p < ∞, x in , and , let and let Note that Lp-Brunn-Minkowski theory: Firey (1961-2); Lutwak (1993-6), LYZ, Vienna, others,…(2000+). E. Lutwak, D. Yang, and G. Zhang, The Brunn-Minkowski-Firey inequality for compact sets,Adv. in Appl. Math. 48 (2012), 407-413.

  5. Motivating Question What’s so special about these binary operations and the few others in convex geometry (Blaschke addition, radial addition, pth radial addition)? We consider binary operations with n ≥ 2 (for this talk!). Here and subscriptso, (o), or s mean “containingo”, “containing o in the interior”, or “o-symmetric”, respectively.

  6. Some Properties Commutativity: Associativity: Homogeneity of degree k: Monotonicity: Identity: Continuity: GL(n) covariance: Projection covariance: Lpaddition has all these properties! Note that for compact convex sets, continuity and GL(n) covariance implies projection covariance.

  7. Theorem 1 [GHW 1] Operations that are projection covariant are precisely those defined for all where M is a 1-unconditional compact convex set in . R.J.G., D. Hug,and W. Weil, Operations between sets in geometry,J. Eur. Math. Soc. (JEMS) 15 (2013), 2297-2353. [GHW1]

  8. Outline of Proof First show where f is homogeneous (of degree 1). By projection covariance, Let Show

  9. Outline of Proof II Then for all Finally, take Let If 1-unconditional M?

  10. Wolfgang Weil 6 April 1945 – 8 February 2018

  11. Pearson’s Theorem Let be a continuous homogeneous (of degree 1) function satisfying the associativity equation Then eitherf(s,t) = 0, or f(s,t) = s, orf(s,t) = t, or f(s,t) = max{s,t}, orf(s,t) = min{s,t}, or there exists p > 0 such that , or there exists a p < 0 such that K. R. Pearson, Interval semirings on R1 with ordinary multiplication, J. Austral. Math. Soc. 6 (1966), 273-288.

  12. Application For r, s, t ≥ 0 , let Then Similarly, other properties transfer from * to hM, for example, continuity.

  13. Theorem 2 [GHW1]: Lp Addition Operations that are associative and projection covariant are precisely those defined for all by where In other words, Corollary 3. An operation is associative, continuous, GL(n) covariant, and has the identity property if and only if it is Lp addition for some 1 ≤ p ≤ ∞.

  14. M-Addition and M-Combination I For 1-unconditional M in , M-addition was introduced by Protasov, who showed that then As a corollary of much more general results, we show in [GHW 1] that when M is 1-unconditional, For an arbitrary set M in , define the M-combination of arbitrary sets K1,…, Km in by V. Y. Protasov, A generalized joint spectral radius. A geometric approach (in Russian), Izv. Ross. Akad. Nauk Ser. Mat. 61 (1997), 99-136; translation in Izv. Math. 61 (1997), 995-1030.

  15. Projects Continue the systematic study of M-addition begun in [GHW1] and extended in T. Mesikepp, M-addition, J. Math. Anal. Appl. 443 (2016), 146-177. 1. (Begun with Jesus Yepes Nicolás.) Study M-subtraction, summands, and decomposition. A. R. Martínez Fernández, E. Saorín Gomez, and J. Yepes Nicolás, p-difference: A counterpart of Minkowski difference in the framework of the Firey-Brunn-Minkowski theory, RACSAM 110 (2016), 613-631. 2. Mesikepp shows that if K1,…, Km and M are convex polytopes, with M contained in a closed orthant of , then is also a convex polytope. Utilize!

  16. M-Addition and M-Combination II • Theorem 1 holds for operations • Projection covariant iffcontinuous and GL(n)covariant also holds for operations • If and M is contained in closed orthant or is 1-unconditional, then • It follows that if is such that its restriction to o-symmetric sets satisfies the hypotheses of Corollary 3, then this restriction must be Lp addition. • If m ≤ n, then if and only if and M is contained in a closed orthant of (The assumption m ≤ n is only needed to conclude that • If , then is a support function whenever if and only if 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]

  17. Milman-Rotem Theorem is monotonic, associative,continuous from below,and has the identity and homothety properties if and only if * = +p for some 1 ≤ p ≤ ∞. homothety: V. Milman and L. Rotem, Characterizing addition of convex sets by polynomiality of volume and by the homothety operation,Commun. Contemp. Math. 17 (2015), no. 3, 1450022, 22 pp. R.J.G. and M. Kiderlen, Operations between functions,Comm. Anal. Geom. 26 (2018), 68 pp., to appear. [GK]

  18. Theorem 4 [GHW 1] An operation (or ) is projection covariant if and only if there is a closed convex set M in such that for all If M = {(1/2,1/2,1/2,1/2)}, then If M = {(0,1,0,1)} (or certain other sets containing this point), then * is Minkowski addition.

  19. Theorem 5 [GHW 1]: Minkowski Addition An operation is projection covariant and has the identity property if and only if it is Minkowski addition. Corollary 6. An operation is continuous, GL(n) covariant, and has the identity property if and only if it is Minkowski addition. Various examples show that none of the assumptions in these results can be omitted.

  20. Some Problems 1. In Theorem 4, can the set M be taken to be a compact convex set in ? (If n ≥ 4, yes.) 2. If then the right-hand side of is a support function and then Otherwise the role of M-addition is unclear. 3. Find the right analogue of Corollary 3 for operations . Note that the operation is projection covariant and associative.

  21. Comments • [Theorem 7.20, GK] considers operations defined by (pointwise operations). It is shown that such operations are associative if and only if F(s,t) = 0, F(s,t) = s, F(s,t) =t, F(s,t)= max{s, 0}, F(s,t) = max{0, t}, F(s,t)= s+t, or, for some 1< p ≤ ∞ and - ∞≤q < 0, • The latter corresponds to for certain all of which represent extensions of Lpaddition to The case q = 0 was suggested in [GHW1], but is just standard Lpaddition applied to conv(K,{o}) and conv(L,{o}). We think q = - ∞ is best; in fact, this gives the convex hull of the extension proposed by LYZ.

  22. Projection Bodies and Zonoids Projection bodies are precisely the o-symmetric, full-dimensional zonoids.

  23. Zonoids: Theorem 5 [GPS] If n ≥ 3, an operation is projection covariant if and only if K * L = aK + bL, for some a, b ≥ 0 and all (By [Theorem 1, GHW1], * must be M-addition. Then use Hlawka’s inequality for support functions of zonoids.) Corollary 6. If n ≥ 3, an operation is continuous, GL(n) covariant, and has the identity property if and only if it is Minkowski addition. Various examples show that none of the assumptions in Theorem 5 and Corollary 6 can be omitted. R.J.G., L. Parapatits,and F. E. Schuster, A characterization of Blaschke addition,Adv. Math. 254 (2014), 396-418. [GPS]

  24. Blaschke Addition For , let Sn – 1(K # L, . ) = Sn – 1(K, . ) + Sn – 1(L, . ) K # L

  25. Minkowski’s Existence Theorem A nonnegative finite Borel measure μ in Sn-1 is the surface area measure of some convex body in if and only if it is not concentrated on a great subsphere and has its centroid at the origin. So the operation is well defined up to translation and can be restricted to an operation …but, when n ≥ 3, cannot be extended to a continuous operation [GHW1]. Curiously, is not monotonic when n ≥ 3 [GPS]!

  26. Central vs. BlaschkeSymmetral In the plane, Blaschke addition and Minkowski addition coincide, up to translation, but… …not for n ≥ 3!

  27. A Crucial Connection Uses the Minkowski map Zonoidsare closed under Minkowski addition.

  28. The Plan 1. Suppose that . Define 2. Find properties of an operation which force * to be Minkowski addition. 3. Find properties of which force the operation * in to possess the properties in (2).

  29. Theorem 7 [GPS]: Blaschke Addition If n ≥ 3, an operation is uniformly continuous in the Lévy-Prokhorov metric and GL(n) covariant if and only if K * L = aK # bL, for some a, b ≥ 0. For nonnegative finite Borel measures µ and ν in Sn-1, define Then define

  30. Corollary 8 [GPS]: Blaschke Addition If n ≥ 3, an operation is uniformly continuous in the Lévy-Prokhorov metric, GL(n) covariant, and has the limit identity property if and only if it is Blaschke addition. Various examples show that none of the assumptions in Theorem 7 and Corollary 8 can be omitted (nor can the word “uniformly”).

  31. End of Part 1

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