Minkowski valuations on $L^p$-spaces
HTML articles powered by AMS MathViewer
- by Andy Tsang PDF
- Trans. Amer. Math. Soc. 364 (2012), 6159-6186 Request permission
Abstract:
A complete classification is obtained of continuous $\operatorname {GL}(n)$- equivariant Minkowski valuations on $L^p(\mathbb {R}^n,|x|dx)$. As a consequence, a characterization of the moment body for functions is obtained.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
- Semyon Alesker, Valuations on convex sets, non-commutative determinants, and pluripotential theory, Adv. Math. 195 (2005), no. 2, 561–595. MR 2146354, DOI 10.1016/j.aim.2004.08.009
- Semyon Alesker, Theory of valuations on manifolds. II, Adv. Math. 207 (2006), no. 1, 420–454. MR 2264077, DOI 10.1016/j.aim.2005.11.015
- Semyon Alesker, Theory of valuations on manifolds: a survey, Geom. Funct. Anal. 17 (2007), no. 4, 1321–1341. MR 2373020, DOI 10.1007/s00039-007-0631-x
- 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
- S. Campi and P. Gronchi, The $L^p$-Busemann-Petty centroid inequality, Adv. Math. 167 (2002), no. 1, 128–141. MR 1901248, DOI 10.1006/aima.2001.2036
- Kai-Seng Chou and Xu-Jia Wang, The $L_p$-Minkowski problem and the Minkowski problem in centroaffine geometry, Adv. Math. 205 (2006), no. 1, 33–83. MR 2254308, DOI 10.1016/j.aim.2005.07.004
- Richard J. Gardner, Geometric tomography, Encyclopedia of Mathematics and its Applications, vol. 58, Cambridge University Press, Cambridge, 1995. MR 1356221
- R. J. Gardner and A. A. Giannopoulos, $p$-cross-section bodies, Indiana Univ. Math. J. 48 (1999), no. 2, 593–613. MR 1722809, DOI 10.1512/iumj.1999.48.1689
- Eric Grinberg and Gaoyong Zhang, Convolutions, transforms, and convex bodies, Proc. London Math. Soc. (3) 78 (1999), no. 1, 77–115. MR 1658156, DOI 10.1112/S0024611599001653
- Peter M. Gruber, Convex and discrete geometry, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 336, Springer, Berlin, 2007. MR 2335496
- Christoph Haberl, $L_p$ intersection bodies, Adv. Math. 217 (2008), no. 6, 2599–2624. MR 2397461, DOI 10.1016/j.aim.2007.11.013
- 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 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
- H. Hadwiger, Vorlesungen über Inhalt, Oberfläche und Isoperimetrie, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1957 (German). MR 0102775
- 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, 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, Invariant valuations on star-shaped sets, Adv. Math. 125 (1997), no. 1, 95–113. MR 1427802, DOI 10.1006/aima.1997.1601
- Daniel A. Klain, Star valuations and dual mixed volumes, Adv. Math. 121 (1996), no. 1, 80–101. MR 1399604, DOI 10.1006/aima.1996.0048
- Daniel A. Klain and Gian-Carlo Rota, Introduction to geometric probability, Lezioni Lincee. [Lincei Lectures], Cambridge University Press, Cambridge, 1997. MR 1608265
- Ludwig, Monika, Covariance matrices and valuations, preprint.
- 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, Intersection bodies and valuations, Amer. J. Math. 128 (2006), no. 6, 1409–1428. MR 2275906
- 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, Projection bodies and valuations, Adv. Math. 172 (2002), no. 2, 158–168. MR 1942402, DOI 10.1016/S0001-8708(02)00021-X
- Ludwig, Monika, Valuations on Sobolev spaces, preprint.
- Monika Ludwig and Matthias Reitzner, A characterization of affine surface area, Adv. Math. 147 (1999), no. 1, 138–172. MR 1725817, DOI 10.1006/aima.1999.1832
- Erwin Lutwak, Centroid bodies and dual mixed volumes, Proc. London Math. Soc. (3) 60 (1990), no. 2, 365–391. MR 1031458, DOI 10.1112/plms/s3-60.2.365
- Erwin Lutwak, On some affine isoperimetric inequalities, J. Differential Geom. 23 (1986), no. 1, 1–13. MR 840399
- Erwin Lutwak, The Brunn-Minkowski-Firey theory. I. Mixed volumes and the Minkowski problem, J. Differential Geom. 38 (1993), no. 1, 131–150. MR 1231704
- Erwin Lutwak, The Brunn-Minkowski-Firey theory. II. Affine and geominimal surface areas, Adv. Math. 118 (1996), no. 2, 244–294. MR 1378681, DOI 10.1006/aima.1996.0022
- 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, Optimal Sobolev norms and the $L^p$ Minkowski problem, Int. Math. Res. Not. , posted on (2006), Art. ID 62987, 21. MR 2211138, DOI 10.1155/IMRN/2006/62987
- 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
- Erwin Lutwak and Gaoyong Zhang, Blaschke-Santaló inequalities, J. Differential Geom. 47 (1997), no. 1, 1–16. MR 1601426
- Peter McMullen, Valuations and dissections, Handbook of convex geometry, Vol. A, B, North-Holland, Amsterdam, 1993, pp. 933–988. MR 1243000
- Peter McMullen and Rolf Schneider, Valuations on convex bodies, Convexity and its applications, Birkhäuser, Basel, 1983, pp. 170–247. MR 731112
- V. D. Milman and A. Pajor, Isotropic position and inertia ellipsoids and zonoids of the unit ball of a normed $n$-dimensional space, Geometric aspects of functional analysis (1987–88), Lecture Notes in Math., vol. 1376, Springer, Berlin, 1989, pp. 64–104. MR 1008717, DOI 10.1007/BFb0090049
- C. M. Petty, Centroid surfaces, Pacific J. Math. 11 (1961), 1535–1547. MR 133733
- Walter Rudin, Real and complex analysis, 3rd ed., McGraw-Hill Book Co., New York, 1987. MR 924157
- 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, 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
- Franz E. Schuster, Valuations and Busemann-Petty type problems, Adv. Math. 219 (2008), no. 1, 344–368. MR 2435426, DOI 10.1016/j.aim.2008.05.001
- Andy Tsang, Valuations on $L^p$-spaces, Int. Math. Res. Not. IMRN 20 (2010), 3993–4023. MR 2738348, DOI 10.1090/S0002-9947-2012-05681-9
- V. Yaskin and M. Yaskina, Centroid bodies and comparison of volumes, Indiana Univ. Math. J. 55 (2006), no. 3, 1175–1194. MR 2244603, DOI 10.1512/iumj.2006.55.2761
Additional Information
- Andy Tsang
- Affiliation: Department of Mathematics, Polytechnic Institute of New York University, Six MetroTech Center, Brooklyn, New York 11201
- Email: atsang122@verizon.net
- Received by editor(s): May 25, 2010
- Published electronically: June 27, 2012
- Additional Notes: This research was taken from the dissertation submitted to the Faculty of the Polytechnic Institute of New York University in partial fulfillment of the requirements for the degree Doctor of Philosophy (Mathematics), June 2010.
- © Copyright 2012 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 364 (2012), 6159-6186
- MSC (2010): Primary 52A20; Secondary 46E30, 52B45
- DOI: https://doi.org/10.1090/S0002-9947-2012-05681-9
- MathSciNet review: 2965739