Volume difference inequalities

Giannopoulos, Apostolos; Koldobsky, Alexander

doi:10.1090/tran/7173

Volume difference inequalities
HTML articles powered by AMS MathViewer

by Apostolos Giannopoulos and Alexander Koldobsky PDF

Trans. Amer. Math. Soc. 370 (2018), 4351-4372 Request permission

Abstract:

We prove several inequalities estimating the distance between volumes of two bodies in terms of the maximal or minimal difference between areas of sections or projections of these bodies. We also provide extensions in which volume is replaced by an arbitrary measure.

References

Shiri Artstein-Avidan, Apostolos Giannopoulos, and Vitali D. Milman, Asymptotic geometric analysis. Part I, Mathematical Surveys and Monographs, vol. 202, American Mathematical Society, Providence, RI, 2015. MR 3331351, DOI 10.1090/surv/202
K. M. Ball, Isometric problems in $\ell _p$ and sections of convex sets, Ph.D. dissertation, Trinity College, Cambridge (1986).
Keith Ball, Shadows of convex bodies, Trans. Amer. Math. Soc. 327 (1991), no. 2, 891–901. MR 1035998, DOI 10.1090/S0002-9947-1991-1035998-3
Keith Ball, Volume ratios and a reverse isoperimetric inequality, J. London Math. Soc. (2) 44 (1991), no. 2, 351–359. MR 1136445, DOI 10.1112/jlms/s2-44.2.351
I. Bárány and Z. Füredi, Approximation of the sphere by polytopes having few vertices, Proc. Amer. Math. Soc. 102 (1988), no. 3, 651–659. MR 928998, DOI 10.1090/S0002-9939-1988-0928998-8
J. Bourgain, On high-dimensional maximal functions associated to convex bodies, Amer. J. Math. 108 (1986), no. 6, 1467–1476. MR 868898, DOI 10.2307/2374532
J. Bourgain, Geometry of Banach spaces and harmonic analysis, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986) Amer. Math. Soc., Providence, RI, 1987, pp. 871–878. MR 934289
J. Bourgain, On the distribution of polynomials on high-dimensional convex sets, Geometric aspects of functional analysis (1989–90), Lecture Notes in Math., vol. 1469, Springer, Berlin, 1991, pp. 127–137. MR 1122617, DOI 10.1007/BFb0089219
Silouanos Brazitikos, Apostolos Giannopoulos, Petros Valettas, and Beatrice-Helen Vritsiou, Geometry of isotropic convex bodies, Mathematical Surveys and Monographs, vol. 196, American Mathematical Society, Providence, RI, 2014. MR 3185453, DOI 10.1090/surv/196
Giorgos Chasapis, Apostolos Giannopoulos, and Dimitris-Marios Liakopoulos, Estimates for measures of lower dimensional sections of convex bodies, Adv. Math. 306 (2017), 880–904. MR 3581320, DOI 10.1016/j.aim.2016.10.035
Susanna Dann, Grigoris Paouris, and Peter Pivovarov, Bounding marginal densities via affine isoperimetry, Proc. Lond. Math. Soc. (3) 113 (2016), no. 2, 140–162. MR 3534969, DOI 10.1112/plms/pdw026
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, The dual Brunn-Minkowski theory for bounded Borel sets: dual affine quermassintegrals and inequalities, Adv. Math. 216 (2007), no. 1, 358–386. MR 2353261, DOI 10.1016/j.aim.2007.05.018
Apostolos Giannopoulos and Alexander Koldobsky, Variants of the Busemann-Petty problem and of the Shephard problem, Int. Math. Res. Not. IMRN 3 (2017), 921–943. MR 3658155, DOI 10.1093/imrn/rnw046
Apostolos Giannopoulos and Emanuel Milman, $M$-estimates for isotropic convex bodies and their $L_q$-centroid bodies, Geometric aspects of functional analysis, Lecture Notes in Math., vol. 2116, Springer, Cham, 2014, pp. 159–182. MR 3364687, DOI 10.1007/978-3-319-09477-9_{1}3
Eric L. Grinberg, Isoperimetric inequalities and identities for $k$-dimensional cross-sections of convex bodies, Math. Ann. 291 (1991), no. 1, 75–86. MR 1125008, DOI 10.1007/BF01445191
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
Fritz John, Extremum problems with inequalities as subsidiary conditions, Studies and Essays Presented to R. Courant on his 60th Birthday, January 8, 1948, Interscience Publishers, Inc., New York, N. Y., 1948, pp. 187–204. MR 0030135
B. Klartag, On convex perturbations with a bounded isotropic constant, Geom. Funct. Anal. 16 (2006), no. 6, 1274–1290. MR 2276540, DOI 10.1007/s00039-006-0588-1
Alexander Koldobsky, Intersection bodies, positive definite distributions, and the Busemann-Petty problem, Amer. J. Math. 120 (1998), no. 4, 827–840. MR 1637955
Alexander Koldobsky, An application of the Fourier transform to sections of star bodies, Israel J. Math. 106 (1998), 157–164. MR 1656857, DOI 10.1007/BF02773465
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
Alexander Koldobsky, Stability in the Busemann-Petty and Shephard problems, Adv. Math. 228 (2011), no. 4, 2145–2161. MR 2836117, DOI 10.1016/j.aim.2011.06.031
Alexander Koldobsky, A hyperplane inequality for measures of convex bodies in $\Bbb R^n$, $n\leq 4$, Discrete Comput. Geom. 47 (2012), no. 3, 538–547. MR 2891246, DOI 10.1007/s00454-011-9362-8
Alexander Koldobsky and Dan Ma, Stability and slicing inequalities for intersection bodies, Geom. Dedicata 162 (2013), 325–335. MR 3009547, DOI 10.1007/s10711-012-9729-x
A. Koldobsky, Stability and separation in volume comparison problems, Math. Model. Nat. Phenom. 8 (2013), no. 1, 156–169. MR 3022986, DOI 10.1051/mmnp/20138111
Alexander Koldobsky, Slicing inequalities for measures of convex bodies, Adv. Math. 283 (2015), 473–488. MR 3383809, DOI 10.1016/j.aim.2015.07.019
Alexander Koldobsky, Slicing inequalities for subspaces of $L_p$, Proc. Amer. Math. Soc. 144 (2016), no. 2, 787–795. MR 3430854, DOI 10.1090/proc12708
Alexander Koldobsky, Stability inequalities for projections of convex bodies, Discrete Comput. Geom. 57 (2017), no. 1, 152–163. MR 3589060, DOI 10.1007/s00454-016-9844-9
Alexander Koldobsky and Alain Pajor, A remark on measures of sections of $L_p$-balls, Geometric aspects of functional analysis, Lecture Notes in Math., vol. 2169, Springer, Cham, 2017, pp. 213–220. MR 3645124
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
Alexander Koldobsky, Dmitry Ryabogin, and Artem Zvavitch, Projections of convex bodies and the Fourier transform, Israel J. Math. 139 (2004), 361–380. MR 2041799, DOI 10.1007/BF02787557
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
Emanuel Milman, Dual mixed volumes and the slicing problem, Adv. Math. 207 (2006), no. 2, 566–598. MR 2271017, DOI 10.1016/j.aim.2005.09.008
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, Projection bodies, Proc. Colloquium on Convexity (Copenhagen, 1965) Kobenhavns Univ. Mat. Inst., Copenhagen, 1967, pp. 234–241. MR 0216369
Richard E. Pfiefer, Maximum and minimum sets for some geometric mean values, J. Theoret. Probab. 3 (1990), no. 2, 169–179. MR 1046328, DOI 10.1007/BF01045156
Rolf Schneider, Zur einem Problem von Shephard über die Projektionen konvexer Körper, Math. Z. 101 (1967), 71–82 (German). MR 218976, DOI 10.1007/BF01135693
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
Gaoyong Zhang, Sections of convex bodies, Amer. J. Math. 118 (1996), no. 2, 319–340. MR 1385280