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Ellipsoids and matrix-valued valuations. (English) Zbl 1033.52012

Author’s abstract: The author obtains a classification of Borel measurable, GL(\(n\)) covariant, symmetric-matrix-valued valuations on the space of \(n\)-dimensional convex polytopes. The only ones turn out to be the moment matrix corresponding to the classical Legendre ellipsoid and the matrix corresponding to the ellipsoid recently discovered by E. Lutwak, D. Yang and G. Zhang.

MSC:

52B45 Dissections and valuations (Hilbert’s third problem, etc.)
52B11 \(n\)-dimensional polytopes
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[1] S. Alesker, Continuous rotation invariant valuations on convex sets , Ann. of Math. (2) 149 (1999), 977–1005. · Zbl 0941.52002 · doi:10.2307/121078
[2] –. –. –. –., Description of continuous isometry covariant valuations on convex sets , Geom. Dedicata 74 (1999), 241–248. · Zbl 0935.52006 · doi:10.1023/A:1005035232264
[3] –. –. –. –., On P. McMullen’s conjecture on translation invariant valuations , Adv. Math. 155 (2000), 239–263. · Zbl 0971.52004 · doi:10.1006/aima.2000.1918
[4] –. –. –. –., Description of translation invariant valuations on convex sets with solution of P. McMullen’s conjecture , Geom. Funct. Anal. 11 (2001), 244–272. · Zbl 0995.52001 · doi:10.1007/PL00001675
[5] R. Gardner, Geometric Tomography , Encyclopedia Math. Appl. 58 , Cambridge Univ. Press, Cambridge, 1995. · Zbl 0864.52001
[6] A. A. Giannopoulos and V. D. Milman,“Euclidean structure in finite dimensional normed spaces” in Handbook of the Geometry of Banach Spaces, Vol. I , ed. W. B. Johnson and J. Lindenstrauss, North-Holland, Amsterdam, 2001, 707–779. · Zbl 1009.46004 · doi:10.1016/S1874-5849(01)80019-X
[7] O. G. Guleryuz, E. Lutwak, D. Yang, and G. Zhang, Information-theoretic inequalities for contoured probability distributions , IEEE Trans. Inform. Theory 48 (2002), 2377–2383. \CMP1 930 297 · Zbl 1062.94530 · doi:10.1109/TIT.2002.800496
[8] H. Hadwiger, Vorlesungen über Inhalt, Oberfläche und Isoperimetrie , Springer, Berlin, 1957. · Zbl 0078.35703
[9] D. A. Klain, Star valuations and dual mixed volumes , Adv. Math. 121 (1996), 80–101. · Zbl 0858.52003 · doi:10.1006/aima.1996.0048
[10] –. –. –. –., Invariant valuations on star-shaped sets , Adv. Math. 125 (1997), 95–113. · Zbl 0889.52007 · doi:10.1006/aima.1997.1601
[11] –. –. –. –., Even valuations on convex bodies , Trans. Amer. Math. Soc. 352 (2000), 71–93. JSTOR: · Zbl 0940.52002 · doi:10.1090/S0002-9947-99-02240-0
[12] D. A. Klain and G.-C. Rota, Introduction to Geometric Probability , Lezioni Lincee (Lincei Lectures), Cambridge Univ. Press, Cambridge, 1997.
[13] K. Leichtweiss, Affine Geometry of Convex Bodies , Barth, Heidelberg, 1998. · Zbl 0374.52002
[14] J. Lindenstrauss and V. D. Milman, “The local theory of normed spaces and its applications to convexity” in Handbook of Convex Geometry, Vol. A, B , ed. P. M. Gruber and J. M. Wills, North-Holland, Amsterdam, 1993, 1149–1220. · Zbl 0791.52003
[15] M. Ludwig, Moment vectors of polytopes , Rend. Circ. Mat. Palermo (2) Suppl. 70 , part 2 (2002), 123–138. \CMP1 962 589 · Zbl 1113.52031
[16] –. –. –. –., Valuations on polytopes containing the origin in their interiors , Adv. Math. 170 (2002), 239–256. \CMP1 932 331 · Zbl 1015.52012 · doi:10.1006/aima.2002.2077
[17] M. Ludwig and M. Reitzner, A characterization of affine surface area , Adv. Math. 147 (1999), 138–172. · Zbl 0947.52003 · doi:10.1006/aima.1999.1832
[18] E. Lutwak, The Brunn-Minkowski-Firey theory, I: Mixed volumes and the Minkowski problem , J. Differential Geom. 38 (1993), 131–150. · Zbl 0788.52007
[19] –. –. –. –., The Brunn-Minkowski-Firey theory, II : Affine and geominimal surface areas, Adv. Math. 118 (1996), 244–294. · Zbl 0853.52005 · doi:10.1006/aima.1996.0022
[20] E. Lutwak and V. Oliker, On the regularity of solutions to a generalization of the Minkowski problem , J. Differential Geom. 41 (1995), 227–246. · Zbl 0867.52003
[21] E. Lutwak, D. Yang, and G. Zhang, \(L_ p\) affine isoperimetric inequalities , J. Differential Geom. 56 (2000), 111–132. · Zbl 1034.52009
[22] –. –. –. –., A new ellipsoid associated with convex bodies , Duke Math. J. 104 (2000), 375–390. · Zbl 0974.52008 · doi:10.1215/S0012-7094-00-10432-2
[23] –. –. –. –., The Cramer-Rao inequality for star bodies , Duke Math. J. 112 (2002), 59–81. \CMP1 890 647 · Zbl 1021.52008 · doi:10.1215/S0012-9074-02-11212-5
[24] E. Lutwak and G. Zhang, Blaschke-Santaló inequalities , J. Differential Geom. 47 (1997), 1–16. · Zbl 0906.52003
[25] P. McMullen, “Valuations and dissections” in Handbook of Convex Geometry, Vol. B , ed. P. M. Gruber and J. Wills, North-Holland, Amsterdam, 1993, 933–988. · Zbl 0791.52014
[26] P. McMullen and R. Schneider, “Valuations on convex bodies” in Convexity and Its Applications , ed. P. M. Gruber and J. Wills, Birkhäuser, Basel, 1983, 170–247. · Zbl 0534.52001
[27] V. D. Milman and A. Pajor, “Isotropic position and inertia ellipsoids and zonoids of the unit ball of a normed \(n\)-dimensional space” in Geometric Aspects of Functional Analysis ( 1987–88.) , ed. J. Lindenstrauss and V. D. Milman, Lecture Notes in Math. 1376 , Springer, Berlin, 1989, 64–104. · Zbl 0679.46012 · doi:10.1007/BFb0090049
[28] V. D. Milman and G. Schechtman, Asymptotic Theory of Finite-Dimensional Normed Spaces , with an appendix by M. Gromov, Lecture Notes in Math. 1200 , Springer, Berlin, 1986.
[29] G. Pisier, The Volume of Convex Bodies and Banach Space Geometry , Cambridge Tracts Math. 94 , Cambridge Univ. Press, Cambridge, 1989. · Zbl 0698.46008
[30] R. Schneider, Convex Bodies: The Brunn-Minkowski Theory , Encyclopedia Math. Appl. 44 , Cambridge Univ. Press, Cambridge, 1993. · Zbl 0798.52001
[31] N. Tomczak-Jaegermann, Banach-Mazur Distances and Finite-Dimensional Operator Ideals , Pitman Monogr. Surveys Pure Appl. Math. 38 , Wiley, New York, 1989. · Zbl 0721.46004
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