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ISSN 1088-6850(e) ISSN 0002-9947(p)

Minkowski valuations

Author(s): Monika Ludwig
Journal: Trans. Amer. Math. Soc. 357 (2005), 4191-4213.
MSC (2000): Primary 52A20; Secondary 52B11, 52B45
Posted: October 28, 2004
MathSciNet review: 2159706
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Abstract: Centroid and difference bodies define $\operatorname{SL}(n)$ equivariant operators on convex bodies and these operators are valuations with respect to Minkowski addition. We derive a classification of $\operatorname{SL}(n)$ equivariant Minkowski valuations and give a characterization of these operators. We also derive a classification of $\operatorname{SL}(n)$ contravariant Minkowski valuations and of -Minkowski valuations.

References:

1.: S. Alesker, Continuous rotation invariant valuations on convex sets, Ann. of Math. (2) 149 (1999), 977-1005. MR 1709308 (2000i:52019)
2.: S. Alesker, On P. McMullen's conjecture on translation invariant valuations, Adv. Math. 155 (2000), 239-263. MR 1794712 (2001k:52013)
3.: S. Alesker, Description of translation invariant valuations on convex sets with solution of P. McMullen's conjecture, Geom. Funct. Anal. 11 (2001), 244-272. MR 1837364 (2002e:52015)
4.: S. Campi and P. Gronchi, The $L\sp p$ -Busemann-Petty centroid inequality, Adv. Math. 167 (2002), 128-141.MR 1901248 (2003e:52011)
5.: R. Gardner, Geometric tomography, Cambridge University Press, Cambridge, 1995.MR 1356221 (96j:52006)
6.: R. Gardner and A. Giannopoulos, -cross-section bodies, Indiana Univ. Math. J. 48 (1999), 593-613. MR 1722809 (2000i:52002)
7.: R. Gardner and G. Zhang, Affine inequalities and radial mean bodies, Amer. J. Math. 120 (1998), 505-528. MR 1623396 (99e:52006)
8.: E. Grinberg and G. Zhang, Convolutions, transforms, and convex bodies, Proc. London Math. Soc. (3) 78 (1999), 77-115.MR 1658156 (99m:52009)
9.: H. Hadwiger, Vorlesungen über Inhalt, Oberfläche und Isoperimetrie, Springer, Berlin, 1957. MR 0102775 (21:1561)
10.: M. Kiderlen, Blaschke and Minkowski endomorphisms of convex bodies, preprint.
11.: D. Klain, A short proof of Hadwiger's characterization theorem, Mathematika 42 (1995), 329-339. MR 1376731 (97e:52008)
12.: D. Klain, Star valuations and dual mixed volumes, Adv. Math. 121 (1996), 80-101. MR 1399604 (97i:52009)
13.: D. Klain, Invariant valuations on star-shaped sets, Adv. Math. 125 (1997), 95-113. MR 1427802 (98a:52009)
14.: D. Klain, Even valuations on convex bodies, Trans. Amer. Math. Soc. 352 (2000), 71-93. MR 1487620 (2000c:52003)
15.: D. Klain and G. Rota, Introduction to geometric probability, Cambridge University Press, Cambridge, 1997. MR 1608265 (2001f:52009)
16.: K. Leichtweiß, Affine geometry of convex bodies, Johann Ambrosius Barth, Heidelberg, 1998. MR 1630116 (2000j:52005)
17.: M. Ludwig, Projection bodies and valuations, Adv. Math. 172 (2002), 158-168. MR 1942402 (2003j:52012)
18.: M. Ludwig, Ellipsoids and matrix valued valuations, Duke Math. J. 119 (2003), 159-188. MR 1991649 (2004e:52015)
19.: M. Ludwig and M. Reitzner, A characterization of affine surface area, Adv. Math. 147 (1999), 138-172. MR 1725817 (2000j:52018)
20.: E. Lutwak, On some affine isoperimetric inequalities, J. Differential Geom. 23 (1986), 1-13. MR 0840399 (87k:52030)
21.: E. Lutwak, Centroid bodies and dual mixed volumes, Proc. Lond. Math. Soc. 60 (1990), 365-391. MR 1031458 (90k:52024)
22.: E. Lutwak, The Brunn-Minkowski-Firey theory. I. Mixed volumes and the Minkowski problem, J. Differential Geom. 38 (1993), 131-150.MR 1231704 (94g:52008)
23.: E. Lutwak, Selected affine isoperimetric inequalities, Handbook of Convex Geometry, Vol. A (P.M. Gruber and J.M. Wills, eds.), North-Holland, Amsterdam, 1993, 151-176. MR 1242979 (94h:52014)
24.: E. Lutwak, The Brunn-Minkowski-Firey theory. II: Affine and geominimal surface areas, Adv. Math. 118 (1996), 244-294.MR 1378681 (97f:52014)
25.: E. Lutwak, D. Yang, and G. Zhang, ${L}\sb p$ affine isoperimetric inequalities, J. Differential Geom. 56 (2000), 111-132. MR 1863023 (2002h:52011)
26.: E. Lutwak, D. Yang, and G. Zhang, Sharp affine ${L}_p$ Sobolev inequalities, J. Differential Geom. 62 (2002), 17-38. MR 1987375 (2004d:46039)
27.: E. Lutwak and G. Zhang, Blaschke-Santaló inequalities, J. Differential Geom. 47 (1997), 1-16. MR 1601426 (2000c:52011)
28.: P. McMullen, Valuations and Euler-type relations on certain classes of convex polytopes, Proc. London Math. Soc. (3) 35 (1977), 113-135.MR 0448239 (56:6548)
29.: P. McMullen, Valuations and dissections, Handbook of Convex Geometry, Vol. B (P.M. Gruber and J.M. Wills, eds.), North-Holland, Amsterdam, 1993, 933-990. MR 1243000 (95f:52018)
30.: P. McMullen and R. Schneider, Valuations on convex bodies, Convexity and its applications (P.M. Gruber and J.M. Wills, eds.), Birkhäuser, 1983, 170-247. MR 0731112 (85e:52001)
31.: V. D. Milman and A. Pajor, Isotropic position and inertia ellipsoids and zonoids of the unit ball of a normed -dimensional space, Geometric aspects of functional analysis (1987-88), Lecture Notes in Math., vol. 1376, Springer, Berlin, 1989, 64-104. MR 1008717 (90g:52003)
32.: C. M. Petty, Centroid surfaces, Pacific J. Math. 11 (1961), 1535-1547.MR 0133733 (24:A3558)
33.: C. M. Petty, Isoperimetric problems, Proceedings of the Conference on Convexity and Combinatorial Geometry (Univ. Oklahoma, Norman, Okla., 1971), Dept. Math., Univ. Oklahoma, Norman, Okla., 1971, 26-41. MR 0362057 (50:14499)
34.: C. A. Rogers and G. C. Shephard, The difference body of a convex body, Arch. Math. 8 (1957), 220-233. MR 0092172 (19:1073f)
35.: R. Schneider, Equivariant endomorphisms of the space of convex bodies, Trans. Amer. Math. Soc. 194 (1974), 53-78. MR 0353147 (50:5633)
36.: R. Schneider, Convex bodies: the Brunn-Minkowski theory, Cambridge Univ. Press, Cambridge, 1993. MR 1216521 (94d:52007)
37.: R. Schneider, Simple valuations on convex bodies, Mathematika 43 (1996), 32-39. MR 1401706 (97f:52016)
38.: G. Zhang, Restricted chord projection and affine inequalities, Geom. Dedicata 39 (1991), 213-222. MR 1119653 (92f:52017)
39.: G. Zhang, The affine Sobolev inequality, J. Differential Geom. 53 (1999), 183-202. MR 1776095 (2001m:53136)

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