𝐺𝐿(𝑛) contravariant Minkowski valuations

Schuster, Franz; Wannerer, Thomas

doi:10.1090/S0002-9947-2011-05364-X

$\mathrm {GL}(n)$ contravariant Minkowski valuations
HTML articles powered by AMS MathViewer

by Franz E. Schuster and Thomas Wannerer PDF

Trans. Amer. Math. Soc. 364 (2012), 815-826 Request permission

Abstract:

A complete classification of all continuous $\mathrm {GL}(n)$ contravariant Minkowski valuations is established. As an application we present a family of sharp isoperimetric inequalities for such valuations which generalize the classical Petty projection inequality.

References

S. Alesker, Continuous rotation invariant valuations on convex sets, Ann. of Math. (2) 149 (1999), no. 3, 977–1005. MR 1709308, DOI 10.2307/121078
S. Alesker, Description of translation invariant valuations on convex sets with solution of P. McMullen’s conjecture, Geom. Funct. Anal. 11 (2001), no. 2, 244–272. MR 1837364, DOI 10.1007/PL00001675
Andreas Bernig, Valuations with Crofton formula and Finsler geometry, Adv. Math. 210 (2007), no. 2, 733–753. MR 2303237, DOI 10.1016/j.aim.2006.07.009
Andreas Bernig and Ludwig Bröcker, Valuations on manifolds and Rumin cohomology, J. Differential Geom. 75 (2007), no. 3, 433–457. MR 2301452
A. Bernig and J.H.G. Fu, Hermitian integral geometry, Ann. of Math. (2), in press.
William J. Firey, Polar means of convex bodies and a dual to the Brunn-Minkowski theorem, Canadian J. Math. 13 (1961), 444–453. MR 177350, DOI 10.4153/CJM-1961-037-0
Joseph H. G. Fu, Structure of the unitary valuation algebra, J. Differential Geom. 72 (2006), no. 3, 509–533. MR 2219942
Richard J. Gardner, Geometric tomography, 2nd ed., Encyclopedia of Mathematics and its Applications, vol. 58, Cambridge University Press, New York, 2006. MR 2251886, DOI 10.1017/CBO9781107341029
Christoph Haberl, Star body valued valuations, Indiana Univ. Math. J. 58 (2009), no. 5, 2253–2276. MR 2583498, DOI 10.1512/iumj.2009.58.3685
C. Haberl, Blaschke valuations, Amer. J. Math., in press.
Christoph Haberl and Monika Ludwig, A characterization of $L_p$ intersection bodies, Int. Math. Res. Not. , posted on (2006), Art. ID 10548, 29. MR 2250020, DOI 10.1155/IMRN/2006/10548
Christoph Haberl and Franz E. Schuster, General $L_p$ affine isoperimetric inequalities, J. Differential Geom. 83 (2009), no. 1, 1–26. MR 2545028
Christoph Haberl and Franz E. Schuster, Asymmetric affine $L_p$ Sobolev inequalities, J. Funct. Anal. 257 (2009), no. 3, 641–658. MR 2530600, DOI 10.1016/j.jfa.2009.04.009
Markus Kiderlen, Blaschke- and Minkowski-endomorphisms of convex bodies, Trans. Amer. Math. Soc. 358 (2006), no. 12, 5539–5564. MR 2238926, DOI 10.1090/S0002-9947-06-03914-6
Daniel A. Klain, An Euler relation for valuations on polytopes, Adv. Math. 147 (1999), no. 1, 1–34. MR 1725813, DOI 10.1006/aima.1999.1831
Daniel A. Klain, Even valuations on convex bodies, Trans. Amer. Math. Soc. 352 (2000), no. 1, 71–93. MR 1487620, DOI 10.1090/S0002-9947-99-02240-0
Daniel A. Klain and Gian-Carlo Rota, Introduction to geometric probability, Lezioni Lincee. [Lincei Lectures], Cambridge University Press, Cambridge, 1997. MR 1608265
Monika Ludwig, Projection bodies and valuations, Adv. Math. 172 (2002), no. 2, 158–168. MR 1942402, DOI 10.1016/S0001-8708(02)00021-X
Monika Ludwig, Ellipsoids and matrix-valued valuations, Duke Math. J. 119 (2003), no. 1, 159–188. MR 1991649, DOI 10.1215/S0012-7094-03-11915-8
Monika Ludwig, Minkowski valuations, Trans. Amer. Math. Soc. 357 (2005), no. 10, 4191–4213. MR 2159706, DOI 10.1090/S0002-9947-04-03666-9
Monika Ludwig, Intersection bodies and valuations, Amer. J. Math. 128 (2006), no. 6, 1409–1428. MR 2275906
M. Ludwig and M. Reitzner, A classification of $\mathrm {SL}(n)$ invariant valuations, Ann. of Math. (2), in press.
Erwin Lutwak, Mixed projection inequalities, Trans. Amer. Math. Soc. 287 (1985), no. 1, 91–105. MR 766208, DOI 10.1090/S0002-9947-1985-0766208-7
Erwin Lutwak, Inequalities for mixed projection bodies, Trans. Amer. Math. Soc. 339 (1993), no. 2, 901–916. MR 1124171, DOI 10.1090/S0002-9947-1993-1124171-8
Erwin Lutwak, Deane Yang, and Gaoyong Zhang, $L_p$ affine isoperimetric inequalities, J. Differential Geom. 56 (2000), no. 1, 111–132. MR 1863023
Erwin Lutwak, Deane Yang, and Gaoyong Zhang, Sharp affine $L_p$ Sobolev inequalities, J. Differential Geom. 62 (2002), no. 1, 17–38. MR 1987375
Erwin Lutwak, Deane Yang, and Gaoyong Zhang, Volume inequalities for subspaces of $L_p$, J. Differential Geom. 68 (2004), no. 1, 159–184. MR 2152912
E. Lutwak, D. Yang, and G. Zhang, Orlicz projection bodies, Adv. Math. 223 (2010), 220–242.
Peter McMullen, Valuations and dissections, Handbook of convex geometry, Vol. A, B, North-Holland, Amsterdam, 1993, pp. 933–988. MR 1243000
C. M. Petty, Projection bodies, Proc. Colloquium on Convexity (Copenhagen, 1965) Kobenhavns Univ. Mat. Inst., Copenhagen, 1967, pp. 234–241. MR 0216369
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, pp. 26–41. MR 0362057
Rolf Schneider, Equivariant endomorphisms of the space of convex bodies, Trans. Amer. Math. Soc. 194 (1974), 53–78. MR 353147, DOI 10.1090/S0002-9947-1974-0353147-1
Rolf Schneider, Convex bodies: the Brunn-Minkowski theory, Encyclopedia of Mathematics and its Applications, vol. 44, Cambridge University Press, Cambridge, 1993. MR 1216521, DOI 10.1017/CBO9780511526282
Rolf Schneider and Franz E. Schuster, Rotation equivariant Minkowski valuations, Int. Math. Res. Not. , posted on (2006), Art. ID 72894, 20. MR 2272092, DOI 10.1155/IMRN/2006/72894
Rolf Schneider and Wolfgang Weil, Zonoids and related topics, Convexity and its applications, Birkhäuser, Basel, 1983, pp. 296–317. MR 731116
Rolf Schneider and Wolfgang Weil, Stochastic and integral geometry, Probability and its Applications (New York), Springer-Verlag, Berlin, 2008. MR 2455326, DOI 10.1007/978-3-540-78859-1
Franz E. Schuster, Convolutions and multiplier transformations of convex bodies, Trans. Amer. Math. Soc. 359 (2007), no. 11, 5567–5591. MR 2327043, DOI 10.1090/S0002-9947-07-04270-5
F.E. Schuster, Crofton measures and Minkowski valuations, Duke Math. J., in press.
Christian Steineder, Subword complexity and projection bodies, Adv. Math. 217 (2008), no. 5, 2377–2400. MR 2388098, DOI 10.1016/j.aim.2007.11.006
A. C. Thompson, Minkowski geometry, Encyclopedia of Mathematics and its Applications, vol. 63, Cambridge University Press, Cambridge, 1996. MR 1406315, DOI 10.1017/CBO9781107325845
Gao Yong Zhang, Restricted chord projection and affine inequalities, Geom. Dedicata 39 (1991), no. 2, 213–222. MR 1119653, DOI 10.1007/BF00182294
Gaoyong Zhang, The affine Sobolev inequality, J. Differential Geom. 53 (1999), no. 1, 183–202. MR 1776095