DUALITY IN CONVEX GEOMETRY…

1. DUALITY IN CONVEX GEOMETRY… WELL, GEOMETRIC TOMOGRAPHY, ACTUALLY! Richard Gardner

2. Scope of Geometric Tomography Computerized tomography Discrete tomography Robot vision Convex geometry Point X-rays Parallel X-rays Imaging Projections; classical Brunn-Minkowski theory Sections through a fixed point; dual Brunn-Minkowski theory ? Integral geometry Pattern recognition Minkowski geometry Stereology and local stereology Local theory of Banach spaces

3. Projections and Sections Theorem 1. (Blaschke and Hessenberg, 1917.) Suppose that 2 ≤ k ≤ n -1, and every k-projectionK|S of a compact convex set in Rn is a k-dimensional ellipsoid. Then K is an ellipsoid. Theorem 1'. (Busemann, 1955.) Suppose that 2 ≤ k ≤ n -1, and every k-sectionK ∩ S of a compact convex set in Rncontaining the origin in its relative interior is a k-dimensional ellipsoid. Then K is an ellipsoid. Busemann’s proof is direct, but Theorem 1' can be obtained from Theorem 1 by using the (not quite trivial) fact that the polar of an ellipsoid containing the origin in its interior is an ellipsoid. Burton, 1976: Theorem 1' holds for arbitrary compact convex sets and hence for arbitrary sets.

4. The SupportFunction

5. The RadialFunction and Star Bodies Star body: Body star-shaped at owhose radial function is positive and continuous, OR one of several other alternative definitions in the literature! E. Lutwak, Dual mixed volumes, Pacific J. Math.58 (1975), 531-538.

6. Polarity – a Projective Notion For a convex body K containing the origin in its interior and u in Sn-1, Then, if S is a subspace, we have We have(φK)* =φ–tK*, for φ in GLn, but… Let K be a convex body in Rn with o in intK, and let φ be a nonsingular projective transformation of Rn, permissible for K, such that o is in intφK. Then there is a nonsingular projective transformationψ, permissible for K*, such that (φK)* = ψK*. P. McMullen and G. C. Shephard, Convex polytopes and the upper bound conjecture,Cambridge University Press, Cambridge, 1971.

7. Polarity Usually Fails Theorem 2. (Süss, Nakajima, 1932.) Suppose that 2 ≤ k ≤ n -1 and that K and L are compact convex sets in Rn. If all k-projectionsK|S and L|S of K and L are homothetic (or translates), then K and L are homothetic (or translates, respectively). Theorem 2'. (Rogers, 1965.) Suppose that 2 ≤ k ≤ n -1 and that K and L are compact convex sets in Rncontaining the origin in their relative interiors. If all k-sectionsK ∩ S and L ∩ S of K and L are homothetic (or translates), then K and L are homothetic (or translates, respectively). Problem 1. (G, Problem 7.1.) Does Theorem 2' hold for star bodies, or perhaps more generally still?

8. Width and Brightness width function brightness function

9. Cauchy’s projection formula Aleksandrov’s Projection Theorem • For o-symmetricconvex bodies K and L, Cosine transform of surface area measure of K

10. Funk-Minkowski Section Theorem section function Spherical Radon transform of(n-1)st power of radial function of K Funk-Minkowski section theorem: For o-symmetricstar bodies K and L,

11. Projection Bodies

12. Shephard’s Problem • Petty,Schneider, 1967: • For o-symmetric convex bodies K and L, • (i) if L is a projection body and (ii) if and only if n = 2. • A counterexample for n = 3

13. Intersection Bodies Erwin Lutwak

14. Busemann-Petty Problem (Star Body Version) • Lutwak,1988: • For o-symmetricstar bodies K and L, • (i) if K is an intersection body and (ii) if and only if n = 2. • Hadwiger,1968: A counterexample for n = 3

15. Minkowskiand Radial Addition

16. Dual Mixed Volumes The dual mixed volume of star bodies K1, K2,…, Knis E. Lutwak, Dual mixed volumes, Pacific J. Math.58 (1975), 531-538. Intrinsic volumes and Kubota’s formula: Dual volumes and the dual Kubota formula (Lutwak, 1979):

17. Characterizations of Projection andIntersection Bodies I Firey, 1965, Lindquist, 1968: A convex body K in Rn is a projection body iff it is a limit (in the Hausdorff metric) of finite Minkowski sums of n-dimensional o-symmetric ellipsoids. Definition: A star body is an intersection body if ρL = Rμ for some finite even Borel measure μ in Sn-1. Goodeyand Weil, 1995: A star body K in Rn is an intersection body iff it is a limit (in the radial metric) of finite radial sums of n-dimensional o-symmetric ellipsoids.

18. Characterizations of Projection andIntersection Bodies II Weil, 1976: Let L be an o-symmetricconvex body in Rn. An o-symmetricconvex body K in Rn is a projection body iff for all o-symmetricconvex bodies M such that ΠL is contained in ΠM. Zhang, 1994: Let L be an o-symmetricstar body in Rn. An o-symmetricstar body K in Rn is an intersection body iff for all o-symmetricstar bodies M such that IL is contained in IM. For a unified treatment, see: P. Goodey, E. Lutwak, and W. Weil,Functional analytic characterizations of classes of convex bodies, Math. Z.222 (1996), 363-381.

19. Lutwak’s Dictionary

20. Is Lutwak’s Dictionary Infallible? Theorem 3. (Goodey, Schneider, and Weil,1997.) Mostconvex bodies in Rn are determined, among all convex bodies, up to translation and reflection in o, by their width and brightness functions. Theorem 3'. (R.J.G., Soranzo, and Volčič, 1999.) The set of star bodies in Rn that are determined, among all star bodies, up to reflection in o, by their k-section functions for all k, is nowhere dense. Notice that this (very rare) phenomenon does not apply within the class of o-symmetric bodies. There I have no example of this sort.

21. Two Unnatural Problems 1. (Busemann, Petty, 1956.) If K and L are o-symmetricconvex bodies in Rnsuch that sK(u) ≤ sL(u) for all u in Sn-1, is V(K) ≤ V(L)? Solved: Yes, n = 2 (trivial), n = 3 (R.J.G., 1994), n = 4 (Zhang,1999). No,n ≥ 5 (Papadimitrakis, 1992, R.J.G., Zhang, 1994). Unified: R.J.G., Koldobsky, and Schlumprecht, 1999. Theorem 4. (Aleksandrov, 1937.) If K and L are o-symmetric convex bodies in R3 whose projections onto every plane have equal perimeters, then K =L. 2. Problem 2. (R.J.G., 1995; G, Problem 7.6.) If K and L are o-symmetricstar bodies in R3 whose sections by every plane through o have equal perimeters, is K =L? Yes. R.J.G. and Volčič, 1994. Unsolved.

22. Aleksandrov-FenchelInequality If K1, K2,…, Knare compact convex sets in Rn, then Dual Aleksandrov-Fenchel Inequality If K1, K2,…, Knare star bodies in Rn, then with equality when if and only if K1, K2,…, Kiare dilatates of each other. Proof of dual Aleksandrov-Fenchel inequality follows directly from an extension of Hölder’s inequality.

23. Brunn-Minkowskiinequality If K and L are convex bodies in Rn, then (B-M) and (M1) with equality iff K and L are homothetic. Dual Brunn-Minkowski inequality If K and L are star bodies in Rn, then (d.B-M) and (d.M1) with equality iff K and L are dilatates.

24. Relations Suppose that H. Groemer,On an inequality of Minkowski for mixed volumes, Geom. Dedicata.33 (1990), 117-122. (B-M)→(M1) R.J.G. and S. Vassallo, J. Math. Anal. Appl. 231 (1999), 568-587 and 245 (2000), 502-512. (d.B-M)→(d.M1)

25. Busemann Intersection Inequality If K is a convex body in Rn containing the origin in its interior, then with equality if and only if K is an o-symmetric ellipsoid. H. Busemann, Volume in terms of concurrent cross-sections, Pacific J. Math.3 (1953), 1-12.

26. Generalized BusemannIntersection Inequality If K is a bounded Borel set in Rn and 1 ≤ i ≤ n, then with equality when 1 < i < n if and only if K is an o-symmetric ellipsoid and when i = 1 if and only if K is an o-symmetric star body, modulo sets of measure zero. H. Busemann and E. Straus, 1960; E. Grinberg, 1991; R. E. Pfiefer, 1990; R.J.G., Vedel Jensen, and Volčič, 2003. R.J.G., The dual Brunn-Minkowski theory for bounded Borel sets: Dual affine quermassintegrals and inequalities, Adv. Math.216 (2007), 358-386.

27. Petty’s ConjecturedProjection Inequality C. Petty, 1971. Let K be a convex body in Rn. Is it true that with equality if and only if K is an ellipsoid? PettyProjectionInequality:

28. Other Remarks • “Bridges”: Polar projection bodies, centroid bodies, p-cosine transform, Fourier transform,… • The Lp-Brunn-Minkowski theory and beyond… • Gauss measure, p-capacity, etc. • Valuation theory… • There may be one all-encompassing duality, but perhaps more likely, several overlapping dualities, each providing part of the picture.