Log-concavity properties of Minkowski valuations
HTML articles powered by AMS MathViewer
- by Astrid Berg, Lukas Parapatits, Franz E. Schuster and Manuel Weberndorfer; With an appendix by Semyon Alesker PDF
- Trans. Amer. Math. Soc. 370 (2018), 5245-5277 Request permission
Abstract:
New Orlicz Brunn–Minkowski inequalities are established for rigid motion compatible Minkowski valuations of arbitrary degree. These extend classical log-concavity properties of intrinsic volumes and generalize seminal results of Lutwak and others. Two different approaches which refine previously employed techniques are explored. It is shown that both lead to the same class of Minkowski valuations for which these inequalities hold. An appendix by Semyon Alesker contains the proof of a new description of generalized translation invariant valuations.References
- Judit Abardia, Difference bodies in complex vector spaces, J. Funct. Anal. 263 (2012), no. 11, 3588–3603. MR 2984076, DOI 10.1016/j.jfa.2012.09.002
- Judit Abardia and Andreas Bernig, Projection bodies in complex vector spaces, Adv. Math. 227 (2011), no. 2, 830–846. MR 2793024, DOI 10.1016/j.aim.2011.02.013
- 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, Hard Lefschetz theorem for valuations, complex integral geometry, and unitarily invariant valuations, J. Differential Geom. 63 (2003), no. 1, 63–95. MR 2015260
- S. Alesker, The multiplicative structure on continuous polynomial valuations, Geom. Funct. Anal. 14 (2004), no. 1, 1–26. MR 2053598, DOI 10.1007/s00039-004-0450-2
- S. Alesker, Hard Lefschetz theorem for valuations and related questions of integral geometry, Geometric aspects of functional analysis, Lecture Notes in Math., vol. 1850, Springer, Berlin, 2004, pp. 9–20. MR 2087146, DOI 10.1007/978-3-540-44489-3_{2}
- Semyon Alesker, A Fourier-type transform on translation-invariant valuations on convex sets, Israel J. Math. 181 (2011), 189–294. MR 2773042, DOI 10.1007/s11856-011-0008-6
- Semyon Alesker and Dmitry Faifman, Convex valuations invariant under the Lorentz group, J. Differential Geom. 98 (2014), no. 2, 183–236. MR 3238311
- Semyon Alesker, Andreas Bernig, and Franz E. Schuster, Harmonic analysis of translation invariant valuations, Geom. Funct. Anal. 21 (2011), no. 4, 751–773. MR 2827009, DOI 10.1007/s00039-011-0125-8
- 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
- Andreas Bernig and Joseph H. G. Fu, Hermitian integral geometry, Ann. of Math. (2) 173 (2011), no. 2, 907–945. MR 2776365, DOI 10.4007/annals.2011.173.2.7
- Christian Berg, Corps convexes et potentiels sphériques, Mat.-Fys. Medd. Danske Vid. Selsk. 37 (1969), no. 6, 64 pp. (1969) (French). MR 254789
- W. Casselman, Canonical extensions of Harish-Chandra modules to representations of $G$, Canad. J. Math. 41 (1989), no. 3, 385–438. MR 1013462, DOI 10.4153/CJM-1989-019-5
- Joseph H. G. Fu, Structure of the unitary valuation algebra, J. Differential Geom. 72 (2006), no. 3, 509–533. MR 2219942
- R. J. Gardner, The Brunn-Minkowski inequality, Bull. Amer. Math. Soc. (N.S.) 39 (2002), no. 3, 355–405. MR 1898210, DOI 10.1090/S0273-0979-02-00941-2
- Richard J. Gardner, Daniel Hug, and Wolfgang Weil, The Orlicz-Brunn-Minkowski theory: a general framework, additions, and inequalities, J. Differential Geom. 97 (2014), no. 3, 427–476. MR 3263511
- Paul Goodey and Wolfgang Weil, The determination of convex bodies from the mean of random sections, Math. Proc. Cambridge Philos. Soc. 112 (1992), no. 2, 419–430. MR 1171176, DOI 10.1017/S0305004100071085
- Paul Goodey and Wolfgang Weil, Sums of sections, surface area measures, and the general Minkowski problem, J. Differential Geom. 97 (2014), no. 3, 477–514. MR 3263512
- H. Groemer, Geometric applications of Fourier series and spherical harmonics, Encyclopedia of Mathematics and its Applications, vol. 61, Cambridge University Press, Cambridge, 1996. MR 1412143, DOI 10.1017/CBO9780511530005
- Michael Grosser, Michael Kunzinger, Michael Oberguggenberger, and Roland Steinbauer, Geometric theory of generalized functions with applications to general relativity, Mathematics and its Applications, vol. 537, Kluwer Academic Publishers, Dordrecht, 2001. MR 1883263, DOI 10.1007/978-94-015-9845-3
- Christoph Haberl, Minkowski valuations intertwining with the special linear group, J. Eur. Math. Soc. (JEMS) 14 (2012), no. 5, 1565–1597. MR 2966660, DOI 10.4171/JEMS/341
- Christoph Haberl, Erwin Lutwak, Deane Yang, and Gaoyong Zhang, The even Orlicz Minkowski problem, Adv. Math. 224 (2010), no. 6, 2485–2510. MR 2652213, DOI 10.1016/j.aim.2010.02.006
- Christoph Haberl and Lukas Parapatits, The centro-affine Hadwiger theorem, J. Amer. Math. Soc. 27 (2014), no. 3, 685–705. MR 3194492, DOI 10.1090/S0894-0347-2014-00781-5
- Christoph Haberl and Franz E. Schuster, General $L_p$ affine isoperimetric inequalities, J. Differential Geom. 83 (2009), no. 1, 1–26. MR 2545028
- 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 and Gian-Carlo Rota, Introduction to geometric probability, Lezioni Lincee. [Lincei Lectures], Cambridge University Press, Cambridge, 1997. MR 1608265
- John M. Lee, Introduction to smooth manifolds, 2nd ed., Graduate Texts in Mathematics, vol. 218, Springer, New York, 2013. MR 2954043
- 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, Minkowski areas and valuations, J. Differential Geom. 86 (2010), no. 1, 133–161. MR 2772547
- Monika Ludwig and Matthias Reitzner, A classification of $\textrm {SL}(n)$ invariant valuations, Ann. of Math. (2) 172 (2010), no. 2, 1219–1267. MR 2680490, DOI 10.4007/annals.2010.172.1223
- Erwin Lutwak, A general isepiphanic inequality, Proc. Amer. Math. Soc. 90 (1984), no. 3, 415–421. MR 728360, DOI 10.1090/S0002-9939-1984-0728360-3
- 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, 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, 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, A new ellipsoid associated with convex bodies, Duke Math. J. 104 (2000), no. 3, 375–390. MR 1781476, DOI 10.1215/S0012-7094-00-10432-2
- Erwin Lutwak, Deane Yang, and Gaoyong Zhang, Orlicz projection bodies, Adv. Math. 223 (2010), no. 1, 220–242. MR 2563216, DOI 10.1016/j.aim.2009.08.002
- Erwin Lutwak, Deane Yang, and Gaoyong Zhang, Orlicz centroid bodies, J. Differential Geom. 84 (2010), no. 2, 365–387. MR 2652465
- P. McMullen, Valuations and Euler-type relations on certain classes of convex polytopes, Proc. London Math. Soc. (3) 35 (1977), no. 1, 113–135. MR 448239, DOI 10.1112/plms/s3-35.1.113
- Mitsuo Morimoto, Analytic functionals on the sphere, Translations of Mathematical Monographs, vol. 178, American Mathematical Society, Providence, RI, 1998. MR 1641900, DOI 10.1090/mmono/178
- Lukas Parapatits, $\textrm {SL}(n)$-contravariant $L_p$-Minkowski valuations, Trans. Amer. Math. Soc. 366 (2014), no. 3, 1195–1211. MR 3145728, DOI 10.1090/S0002-9947-2013-05750-9
- Lukas Parapatits, $\textrm {SL}(n)$-covariant $L_p$-Minkowski valuations, J. Lond. Math. Soc. (2) 89 (2014), no. 2, 397–414. MR 3188625, DOI 10.1112/jlms/jdt068
- Lukas Parapatits and Franz E. Schuster, The Steiner formula for Minkowski valuations, Adv. Math. 230 (2012), no. 3, 978–994. MR 2921168, DOI 10.1016/j.aim.2012.03.024
- Lukas Parapatits and Thomas Wannerer, On the inverse Klain map, Duke Math. J. 162 (2013), no. 11, 1895–1922. MR 3090780, DOI 10.1215/00127094-2333971
- Marko Petkovšek, Herbert S. Wilf, and Doron Zeilberger, $A=B$, A K Peters, Ltd., Wellesley, MA, 1996. With a foreword by Donald E. Knuth; With a separately available computer disk. MR 1379802
- 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, Second expanded edition, Encyclopedia of Mathematics and its Applications, vol. 151, Cambridge University Press, Cambridge, 2014. MR 3155183
- Franz E. Schuster, Volume inequalities and additive maps of convex bodies, Mathematika 53 (2006), no. 2, 211–234 (2007). MR 2343256, DOI 10.1112/S0025579300000103
- 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, Crofton measures and Minkowski valuations, Duke Math. J. 154 (2010), no. 1, 1–30. MR 2668553, DOI 10.1215/00127094-2010-033
- Franz E. Schuster and Thomas Wannerer, $\textrm {GL}(n)$ contravariant Minkowski valuations, Trans. Amer. Math. Soc. 364 (2012), no. 2, 815–826. MR 2846354, DOI 10.1090/S0002-9947-2011-05364-X
- Franz E. Schuster and Thomas Wannerer, Even Minkowski valuations, Amer. J. Math. 137 (2015), no. 6, 1651–1683. MR 3432270, DOI 10.1353/ajm.2015.0041
- F.E. Schuster and T. Wannerer, Minkowski valuations and generalized valuations, arXiv:1507.05412.
- Franz E. Schuster and Manuel Weberndorfer, Volume inequalities for asymmetric Wulff shapes, J. Differential Geom. 92 (2012), no. 2, 263–283. MR 2998673
- Nolan R. Wallach, Harmonic analysis on homogeneous spaces, Pure and Applied Mathematics, No. 19, Marcel Dekker, Inc., New York, 1973. MR 0498996
- Nolan R. Wallach, Real reductive groups. I, Pure and Applied Mathematics, vol. 132, Academic Press, Inc., Boston, MA, 1988. MR 929683
- Tuo Wang, The affine Sobolev-Zhang inequality on $BV(\Bbb R^n)$, Adv. Math. 230 (2012), no. 4-6, 2457–2473. MR 2927377, DOI 10.1016/j.aim.2012.04.022
- Wei Wang, $L_p$ Brunn-Minkowski type inequalities for Blaschke-Minkowski homomorphisms, Geom. Dedicata 164 (2013), 273–285. MR 3054628, DOI 10.1007/s10711-012-9772-7
- Thomas Wannerer, $\textrm {GL}(n)$ equivariant Minkowski valuations, Indiana Univ. Math. J. 60 (2011), no. 5, 1655–1672. MR 2997003, DOI 10.1512/iumj.2011.60.4425
- Thomas Wannerer, The module of unitarily invariant area measures, J. Differential Geom. 96 (2014), no. 1, 141–182. MR 3161388
- Manuel Weberndorfer, Shadow systems of asymmetric $L_p$ zonotopes, Adv. Math. 240 (2013), 613–635. MR 3046320, DOI 10.1016/j.aim.2013.02.022
- Gaoyong Zhang, The affine Sobolev inequality, J. Differential Geom. 53 (1999), no. 1, 183–202. MR 1776095
Additional Information
- Astrid Berg
- Affiliation: Vienna University of Technology, Institute of Discrete Mathematics and Geometry, Wiedner Hauptstrasse 8–10, 1040 Vienna, Austria
- Email: aberg@posteo.net
- Lukas Parapatits
- Affiliation: Department of Mathematics, ETH Zurich, Rämistrasse 101, 8092 Zurich, Switzerland
- MR Author ID: 979076
- Email: lukas.parapatits@math.ethz.ch
- Franz E. Schuster
- Affiliation: Vienna University of Technology, Institute of Discrete Mathematics and Geometry, Wiedner Hauptstrasse 8–10, 1040 Vienna, Austria
- MR Author ID: 764916
- Email: franz.schuster@tuwien.ac.at
- Manuel Weberndorfer
- Affiliation: Vienna University of Technology, Institute of Discrete Mathematics and Geometry, Wiedner Hauptstrasse 8–10, 1040 Vienna, Austria
- MR Author ID: 1001498
- Email: m.weberndorfer@gmail.com
- Semyon Alesker
- Affiliation: Department of Mathematics, Tel Aviv University, Ramat Aviv, 69978 Tel Aviv, Israel
- MR Author ID: 367436
- Received by editor(s): September 2, 2015
- Received by editor(s) in revised form: February 11, 2017
- Published electronically: March 21, 2018
- Additional Notes: The work of the first, second, and third authors was supported by the European Research Council (ERC) within the project “Isoperimetric Inequalities and Integral Geometry”, Project number: 306445. The second author was also supported by the ETH Zurich Postdoctoral Fellowship Program and the Marie Curie Actions for People COFUND Program.
- © Copyright 2018 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 370 (2018), 5245-5277
- MSC (2010): Primary 52A38, 52B45
- DOI: https://doi.org/10.1090/tran/7434
- MathSciNet review: 3787383