Asymmetric $L_p$-difference bodies
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- by Weidong Wang and Tongyi Ma PDF
- Proc. Amer. Math. Soc. 142 (2014), 2517-2527 Request permission
Abstract:
Lutwak introduced the $L_p$-difference body of a convex body as the Firey $L_p$-combination of the body and its reflection at the origin. In this paper, we define the notion of asymmetric $L_p$-difference bodies and study some of their properties. In particular, we determine the extremal values of the volumes of asymmetric $L_p$-difference bodies and their polars, respectively.References
- 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
- 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
- Chiara Bianchini and Andrea Colesanti, A sharp Rogers and Shephard inequality for the $p$-difference body of planar convex bodies, Proc. Amer. Math. Soc. 136 (2008), no. 7, 2575–2582. MR 2390529, DOI 10.1090/S0002-9939-08-09209-5
- William J. Firey, Mean cross-section measures of harmonic means of convex bodies, Pacific J. Math. 11 (1961), 1263–1266. MR 140003
- Wm. J. Firey, $p$-means of convex bodies, Math. Scand. 10 (1962), 17–24. MR 141003, DOI 10.7146/math.scand.a-10510
- 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, 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 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
- Christoph Haberl, Franz E. Schuster, and Jie Xiao, An asymmetric affine Pólya-Szegö principle, Math. Ann. 352 (2012), no. 3, 517–542. MR 2885586, DOI 10.1007/s00208-011-0640-9
- 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, Valuations in the affine geometry of convex bodies, Integral geometry and convexity, World Sci. Publ., Hackensack, NJ, 2006, pp. 49–65. MR 2240973, DOI 10.1142/9789812774644_{0}005
- Monika Ludwig, Minkowski areas and valuations, J. Differential Geom. 86 (2010), no. 1, 133–161. MR 2772547
- 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, 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
- C. A. Rogers and G. C. Shephard, The difference body of a convex body, Arch. Math. (Basel) 8 (1957), 220–233. MR 92172, DOI 10.1007/BF01899997
- 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
- 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
Additional Information
- Weidong Wang
- Affiliation: Department of Mathematics, China Three Gorges University, Yichang, 443002, People’s Republic of China
- Email: wdwxh722@163.com
- Tongyi Ma
- Affiliation: Department of Mathematics, Hexi University, Gansu Zhangye, 734000, People’s Republic of China
- Email: gsmatongyi@hotmail.com
- Received by editor(s): April 4, 2011
- Received by editor(s) in revised form: August 1, 2011, and July 2, 2012
- Published electronically: March 27, 2014
- Additional Notes: The authors’ research was supported in part by the Natural Science Foundation of China (grants No. 11371224, 11161019) and Science Foundation of China Three Gorges University
- Communicated by: Michael Wolf
- © Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 142 (2014), 2517-2527
- MSC (2010): Primary 52A40, 52A20
- DOI: https://doi.org/10.1090/S0002-9939-2014-11919-8
- MathSciNet review: 3195772