1. Structural Theory of Addition and Symmetrization in Convex Geometry
Richard Gardner
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2. Minkowski Addition
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3. The Brunn-Minkowski Theory
Combine Minkowski addition + and volume V.
Minkowski’s Theorem on Mixed Volumes:
If Ki, i = 1,…, m, are compact convex sets in ,
ti ≥ 0, i = 1,…, m, and
then
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4. Lp Addition and let Note that
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),
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5. Motivating Question We consider binary operations
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 subscripts o, (o), or s mean “containing o”, “containing o in the interior”, or “o-symmetric”, respectively.
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6. Some Properties Commutativity: Associativity: Homogeneity of degree k:
Monotonicity:
Identity:
Continuity:
GL(n) covariance:
Projection covariance:
Lp addition has all these properties!
Note that for compact convex sets, continuity and
GL(n) covariance implies projection covariance.
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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),
[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
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11. Pearson’s Theorem
Let be a continuous homogeneous (of degree 1) function satisfying the associativity equation
Then either f(s,t) = 0, or f(s,t) = s, or f(s,t) = t, or f(s,t) = max{s,t}, or f(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),
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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 ≤ ∞.
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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), ; translation in Izv. Math. 61 (1997),
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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),
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),
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!
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16. M-Addition and M-Combination II
Theorem 1 holds for operations
Projection covariant iff continuous 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),
[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, , 22 pp.
R.J.G. and M. Kiderlen, Operations between functions, Comm. Anal. Geom. 26 (2018), 68 pp., to appear.
[GK]
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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.
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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.
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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.
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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 Lp addition to The case q = 0 was suggested in [GHW1], but is just standard Lp addition 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.
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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),
[GPS]
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24. Blaschke Addition Sn – 1(K # L, . ) = Sn – 1(K, . ) + Sn – 1(L, . )
For , let
Sn – 1(K # L, . ) = Sn – 1(K, . ) + Sn – 1(L, . )
K # L
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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. Blaschke Symmetral
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
Zonoids are closed under Minkowski addition.
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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).
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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
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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”).
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31. End of Part 1
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