Counterexamples to convexity of $k$-intersection bodies
HTML articles powered by AMS MathViewer
- by Vladyslav Yaskin PDF
- Proc. Amer. Math. Soc. 142 (2014), 4355-4363 Request permission
Abstract:
It is a well-known result due to Busemann that the intersection body of an origin-symmetric convex body is also convex. Koldobsky introduced the notion of $k$-intersection bodies. We show that the $k$-intersection body of an origin-symmetric convex body is not necessarily convex if $k>1$.References
- Gautier Berck, Convexity of $L_p$-intersection bodies, Adv. Math. 222 (2009), no. 3, 920–936. MR 2553373, DOI 10.1016/j.aim.2009.05.009
- Jean Bourgain and Gaoyong Zhang, On a generalization of the Busemann-Petty problem, Convex geometric analysis (Berkeley, CA, 1996) Math. Sci. Res. Inst. Publ., vol. 34, Cambridge Univ. Press, Cambridge, 1999, pp. 65–76. MR 1665578, DOI 10.2977/prims/1195144828
- 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
- 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
- R. J. Gardner, A. Koldobsky, and T. Schlumprecht, An analytic solution to the Busemann-Petty problem on sections of convex bodies, Ann. of Math. (2) 149 (1999), no. 2, 691–703. MR 1689343, DOI 10.2307/120978
- I. M. Gel′fand and G. E. Shilov, Generalized functions. Vol. 1, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1964 [1977]. Properties and operations; Translated from the Russian by Eugene Saletan. MR 0435831
- 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 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
- A. Koldobsky, A functional analytic approach to intersection bodies, Geom. Funct. Anal. 10 (2000), no. 6, 1507–1526. MR 1810751, DOI 10.1007/PL00001659
- Alexander Koldobsky, Fourier analysis in convex geometry, Mathematical Surveys and Monographs, vol. 116, American Mathematical Society, Providence, RI, 2005. MR 2132704, DOI 10.1090/surv/116
- A. Koldobsky, H. König, and M. Zymonopoulou, The complex Busemann-Petty problem on sections of convex bodies, Adv. Math. 218 (2008), no. 2, 352–367. MR 2407938, DOI 10.1016/j.aim.2007.12.006
- A. Koldobsky, G. Paouris, and M. Zymonopoulou, Isomorphic properties of intersection bodies, J. Funct. Anal. 261 (2011), no. 9, 2697–2716. MR 2826412, DOI 10.1016/j.jfa.2011.07.011
- A. Koldobsky, G. Paouris, and M. Zymonopoulou, Complex intersection bodies, J. Lond. Math. Soc. (2) 88 (2013), no. 2, 538–562. MR 3106735, DOI 10.1112/jlms/jdt014
- Alexander Koldobsky and Vladyslav Yaskin, The interface between convex geometry and harmonic analysis, CBMS Regional Conference Series in Mathematics, vol. 108, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2008. MR 2365157
- Jaegil Kim, Vladyslav Yaskin, and Artem Zvavitch, The geometry of $p$-convex intersection bodies, Adv. Math. 226 (2011), no. 6, 5320–5337. MR 2775903, DOI 10.1016/j.aim.2011.01.011
- Erwin Lutwak, Intersection bodies and dual mixed volumes, Adv. in Math. 71 (1988), no. 2, 232–261. MR 963487, DOI 10.1016/0001-8708(88)90077-1
- 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 and Gaoyong Zhang, Blaschke-Santaló inequalities, J. Differential Geom. 47 (1997), no. 1, 1–16. MR 1601426
- Emanuel Milman, Generalized intersection bodies, J. Funct. Anal. 240 (2006), no. 2, 530–567. MR 2261694, DOI 10.1016/j.jfa.2006.04.004
- Emanuel Milman, Generalized intersection bodies are not equivalent, Adv. Math. 217 (2008), no. 6, 2822–2840. MR 2397468, DOI 10.1016/j.aim.2007.11.007
- 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
- Jared Schlieper, A note on $k$-intersection bodies, Proc. Amer. Math. Soc. 135 (2007), no. 7, 2081–2088. MR 2299484, DOI 10.1090/S0002-9939-07-08774-6
- Vladyslav Yaskin, On strict inclusions in hierarchies of convex bodies, Proc. Amer. Math. Soc. 136 (2008), no. 9, 3281–3291. MR 2407094, DOI 10.1090/S0002-9939-08-09424-0
- 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
- Gaoyong Zhang, Sections of convex bodies, Amer. J. Math. 118 (1996), no. 2, 319–340. MR 1385280
- Gaoyong Zhang, A positive solution to the Busemann-Petty problem in $\mathbf R^4$, Ann. of Math. (2) 149 (1999), no. 2, 535–543. MR 1689339, DOI 10.2307/120974
Additional Information
- Vladyslav Yaskin
- Affiliation: Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta T6G 2G1, Canada
- MR Author ID: 650371
- Email: vladyaskin@math.ualberta.ca
- Received by editor(s): February 3, 2013
- Published electronically: August 14, 2014
- Additional Notes: This research was supported in part by NSERC
- Communicated by: Alexander Iosevich
- © 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), 4355-4363
- MSC (2010): Primary 52A20
- DOI: https://doi.org/10.1090/S0002-9939-2014-12254-4
- MathSciNet review: 3267003