Abstract
Given \(\gamma >1\) we say that a Borel measure \(\nu \) on \(S^{n-1}\) is a \(\gamma \)-approximation of an isotropic measure if
where \(I_n\) is the identity matrix. We provide a generalization of Barthe’s continuous version of the Brascamp–Lieb inequalities to the context of these approximate isotropic measures, and we apply these inequalities to obtain stability results for some classical positions of convex bodies.
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References
Artstein-Avidan, S., Giannopoulos, A., Milman, V.D.: Asymptotic Geometric Analysis, Part I, Mathematical Surveys and Monographs, vol. 202. American Mathematical Society, Providence (2015)
Ball, K.M.: Volumes of Sections of Cubes and Related Problems. Lecture Notes in Mathematics, vol. 1376. Springer, Berlin (1989)
Ball, K.M.: Volume ratios and a reverse isoperimetric inequality. J. Lond. Math. Soc. (2) 44, 351–359 (1991)
Ball, K.M.: Convex geometry and functional analysis. In: Handbook of the geometry of Banach Spaces, vol. I, pp. 161–194. North-Holland, Amsterdam (2001)
Bapat, R.B.: Mixed discriminants of positive semidefinite matrices. Linear Algebra Appl. 126, 107–124 (1989)
Barthe, F.: Inégalités de Brascamp–Lieb et convexité. C. R. Acad. Sci. Paris Ser. I Math. 324(8), 885–888 (1997)
Barthe, F.: On a reverse form of the Brascamp–Lieb inequality. Invent. Math. 134, 335–361 (1998)
Barthe, F.: A continuous version of the Brascamp–Lieb inequalities. In: Geometric Aspects of Functional Analysis. Lecture Notes in Mathematics, vol. 1850, pp. 53–63. Springer, Berlin (2004)
Bourgain, J., Milman, V.D.: New volume ratio properties for convex symmetric bodies in \({{\mathbb{R}}}^n\). Invent. Math. 88, 319–340 (1987)
Brascamp, H.J., Lieb, E.H.: Best constants in Young’s inequality, its converse and its generalization to more than three functions. Adv. Math. 20, 151–173 (1976)
Brazitikos, S.: Brascamp–Lieb inequality and quantitative versions of Helly’s theorem. Mathematika 63, 272–291 (2017)
Giannopoulos, A., Milman, V.D.: Extremal problems and isotropic positions of convex bodies. Isr. J. Math. 117, 29–60 (2000)
Giannopoulos, A., Papadimitrakis, M.: Isotropic surface area measures. Mathematika 46, 1–13 (1999)
Giannopoulos, A., Milman, V.D., Rudelson, M.: Convex bodies with minimal mean width. In: Geometric Aspects of Functional Analysis. Lecture Notes in Mathematics, vol. 1745, pp. 81–93 (2000)
Gordon, Y., Meyer, M., Reisner, S.: Zonoids with minimal volume product—a new proof. Proc. Am. Math. Soc. 104, 273–276 (1988)
John, F.: Extremum Problems with Inequalities as Subsidiary Conditions. Courant Anniversary Volume, pp. 187–204. Interscience, New York (1948)
Lutwak, E., Yang, D., Zhang, G.: Volume inequalities for subspaces of \(L^p\). J. Differ. Geom. 68, 159–184 (2004)
Lutwak, E., Yang, D., Zhang, G.: A volume inequality for polar bodies. J. Differ. Geom. 84, 163–178 (2010)
Maresch, G., Schuster, F.: The sine transform of isotropic measures. Int. Math. Res. Not. IMRN 4, 717–739 (2012)
Petty, C.M.: Surface area of a convex body under affine transformations. Proc. Am. Math. Soc. 12, 824–828 (1961)
Schneider, R.: Convex Bodies: The Brunn–Minkowski Theory, 2nd expanded edn. Encyclopedia of Mathematics and Its Applications, vol. 151, Cambridge University Press, Cambridge (2014)
Valdimarsson, S.: Geometric Brascamp–Lieb has the optimal best constant. J. Geom. Anal. 21(4), 1036–1043 (2011)
Acknowledgements
We acknowledge support from the Hellenic Foundation for Research & Innovation (H.F.R.I.) under the “First Call for H.F.R.I. Research Projects to support Faculty members and Researchers” (Project Number: 1849). We would like to thank the referee for valuable comments.
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Brazitikos, S., Giannopoulos, A. Continuous Version of the Approximate Geometric Brascamp–Lieb Inequalities. J Geom Anal 32, 174 (2022). https://doi.org/10.1007/s12220-022-00909-z
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DOI: https://doi.org/10.1007/s12220-022-00909-z