Log-concavity properties of Minkowski valuations

Berg, Astrid; Parapatits, Lukas; Schuster, Franz; Weberndorfer, Manuel

doi:10.1090/tran/7434

Log-concavity properties of Minkowski valuations
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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