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2/10/20141 DUALITY IN CONVEX GEOMETRY… Richard Gardner WELL, GEOMETRIC TOMOGRAPHY, ACTUALLY!

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2/10/20142 Scope of Geometric Tomography Point X-rays Integral geometry Local theory of Banach spaces Robot vision Stereology and local stereology Computerized tomography Discrete tomography Parallel X-rays Minkowski geometry Imaging Pattern recognition Convex geometry Sections through a fixed point; dual Brunn-Minkowski theory Projections; classical Brunn- Minkowski theory ?

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2/10/20143 Projections and Sections Theorem 1. (Blaschke and Hessenberg, 1917.) Suppose that 2 k n -1, and every k-projection K|S of a compact convex set in R n is a k-dimensional ellipsoid. Then K is an ellipsoid. Theorem 1'. (Busemann, 1955.) Suppose that 2 k n -1, and every k-section K S of a compact convex set in R n containing the origin in its relative interior is a k-dimensional ellipsoid. Then K is an ellipsoid. Busemanns 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.

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2/10/20144 The Support Function

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2/10/20145 The Radial Function and Star Bodies E. Lutwak, Dual mixed volumes, Pacific J. Math. 58 (1975), 531-538. Star body: Body star-shaped at o whose radial function is positive and continuous, OR one of several other alternative definitions in the literature!

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2/10/20146 Polarity – a Projective Notion Let K be a convex body in R n with o in int K, and let φ be a nonsingular projective transformation of R n, permissible for K, such that o is in int φK. Then there is a nonsingular projective transformation ψ, 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. For a convex body K containing the origin in its interior and u in S n-1, Then, if S is a subspace, we have We have (φK)* = φ –t K*, for φ in GL n, but…

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2/10/20147 Polarity Usually Fails Theorem 2. (Süss, Nakajima, 1932.) Suppose that 2 k n -1 and that K and L are compact convex sets in R n. If all k-projections K|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 R n containing the origin in their relative interiors. If all k-sections K 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?

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2/10/20148 Width and Brightness width function brightness function

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2/10/20149 Aleksandrovs Projection Theorem For o-symmetric convex bodies K and L, Cauchys projection formula Cosine transform of surface area measure of K

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2/10/201410 Funk-Minkowski Section Theorem Funk-Minkowski section theorem: For o-symmetric star bodies K and L, section function Spherical Radon transform of (n-1)st power of radial function of K

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2/10/201411 Projection Bodies

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2/10/201412 Shephards 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

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2/10/201413 Intersection Bodies Erwin Lutwak

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2/10/201414 Busemann-Petty Problem (Star Body Version) Lutwak,1988: For o-symmetric star 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

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2/10/201415 Minkowski and Radial Addition

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2/10/201416 Dual Mixed Volumes The dual mixed volume of star bodies K 1, K 2,…, K n is E. Lutwak, Dual mixed volumes, Pacific J. Math. 58 (1975), 531-538. Intrinsic volumes and Kubotas formula: Dual volumes and the dual Kubota formula (Lutwak, 1979):

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2/10/201417 Characterizations of Projection and Intersection Bodies I Firey, 1965, Lindquist, 1968: A convex body K in R n 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 ρ L = Rμ for some finite even Borel measure μ in S n-1. Goodey and Weil, 1995: A star body K in R n is an intersection body iff it is a limit (in the radial metric) of finite radial sums of n-dimensional o-symmetric ellipsoids.

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2/10/201418 Characterizations of Projection and Intersection Bodies II Weil, 1976: Let L be an o-symmetric convex body in R n. An o- symmetric convex body K in R n is a projection body iff for all o-symmetric convex bodies M such that ΠL is contained in ΠM. For a unified treatment, see: Zhang, 1994: Let L be an o-symmetric star body in R n. An o- symmetric star body K in R n is an intersection body iff for all o-symmetric star bodies M such that IL is contained in IM. P. Goodey, E. Lutwak, and W. Weil, Functional analytic characterizations of classes of convex bodies, Math. Z. 222 (1996), 363-381.

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2/10/201419 Lutwaks Dictionary Convex bodiesStar bodies ProjectionsSections through o Support function h K Radial function ρ K Brightness function b K Section function s K Projection body ΠKIntersection body IK Cosine transformSpherical Radon transform Surface area measureρ K n-1 Mixed volumesDual mixed volumes Brunn-Minkowski ineq.Dual B-M inequality Aleksandrov-FenchelDual A-F inequality

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2/10/201420 Is Lutwaks Dictionary Infallible? Theorem 3. (Goodey, Schneider, and Weil,1997.) Most convex bodies in R n 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 R n 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.

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2/10/201421 Two Unnatural Problems 1. ( Busemann, Petty, 1956.) If K and L are o-symmetric convex bodies in R n such that s K (u) s L (u) for all u in S n-1, is V(K) V(L)? 2. Problem 2. (R.J.G., 1995; G, Problem 7.6.) If K and L are o- symmetric star bodies in R 3 whose sections by every plane through o have equal perimeters, is K =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. Unsolved. Theorem 4. (Aleksandrov, 1937.) If K and L are o-symmetric convex bodies in R 3 whose projections onto every plane have equal perimeters, then K =L. Yes. R.J.G. and Volčič, 1994.

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2/10/201422 Aleksandrov-Fenchel Inequality If K 1, K 2,…, K n are compact convex sets in R n, then Dual Aleksandrov-Fenchel Inequality If K 1, K 2,…, K n are star bodies in R n, then with equality when if and only if K 1, K 2,…, K i are dilatates of each other. Proof of dual Aleksandrov-Fenchel inequality follows directly from an extension of Hölders inequality.

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2/10/201423 Brunn-Minkowski inequality If K and L are convex bodies in R n, then with equality iff K and L are homothetic. and (B-M) (M1) Dual Brunn-Minkowski inequality If K and L are star bodies in R n, then and with equality iff K and L are dilatates. (d.B-M) (d.M1)

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2/10/201424Relations Suppose that (B-M)(M1) (d.B-M)(d.M1) H. Groemer, On an inequality of Minkowski for mixed volumes, Geom. Dedicata. 33 (1990), 117-122. R.J.G. and S. Vassallo, J. Math. Anal. Appl. 231 (1999), 568-587 and 245 (2000), 502-512.

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2/10/201425 Busemann Intersection Inequality with equality if and only if K is an o-symmetric ellipsoid. If K is a convex body in R n containing the origin in its interior, then H. Busemann, Volume in terms of concurrent cross-sections, Pacific J. Math. 3 (1953), 1-12.

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2/10/201426 Generalized Busemann Intersection Inequality 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. If K is a bounded Borel set in R n and 1 i n, then 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.

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2/10/201427 Pettys Conjectured Projection Inequality with equality if and only if K is an ellipsoid? Let K be a convex body in R n. Is it true that C. Petty, 1971. Petty Projection Inequality:

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2/10/201428 Other Remarks Bridges: Polar projection bodies, centroid bodies, p- cosine transform, Fourier transform,… The L p -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.

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