Abstract
We extend the notion of John’s ellipsoid to the setting of integrable log-concave functions. This will allow us to define the integral ratio of a log-concave function, which will extend the notion of volume ratio, and we will find the log-concave function maximizing the integral ratio. A reverse functional affine isoperimetric inequality will be given, written in terms of this integral ratio. This can be viewed as a stability version of the functional affine isoperimetric inequality.
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References
Alonso-Gutiérrez, D.: On a reverse Petty projection inequality for projections of convex bodies. Adv. Geom. 14, 215–223 (2014)
Arstein-Avidan, S., Klartag, B., Schütt, C., Werner, E.: Functional affine-isoperimetry and an inverse logarithmic Sobolev inequality. J. Funct. Anal. 262(9), 4181–4204 (2012)
Arstein, S., Klartag, B., Milman, V.: The Santaló point of a function, and a functional form of Blaschke-Santaló inequality. Mathematika 51, 33–48 (2004)
Ball, K.: Volume ratios and a reverse isoperimetric inequality. J. Lond. Math. Soc. 44(2), 351–359 (1991)
Barthe, F.: An extremal property of the mean width of the simplex. Math. Ann. 310(4), 685–693 (1998)
Bobkov, S.G., Madiman, M.: Reverse Brunn–Minkowski and reverse entropy power inequalities for convex measures. J. Funct. Anal. 262(7), 3309–3339 (2012)
Bobkov, S.G., Madiman, M.: Entropy and the hyperplane conjecture in convex geometry. In: 2010 IEEE International Symposium on Information Theory Proceedings (ISIT), Austin, pp. 1438–1442 (2010)
Brascamp, H.J., Lieb, E.H.: Best constants in Young’s inequality, its converse, and its generalization to mor than three functions. Adv. Math. 20, 151–173 (1976)
Colesanti, A.: Functional inequalities related to the rogers-Shephard inequality. Mathematika 53(1), 81–101 (2006)
Colesanti, A., Fragalà, I.: The area measure of log-concave functions and related inequalities. Adv. Math. 244, 708–749 (2013)
Fradelizi, M., Meyer, M.: Some functional forms of Blaschke-Santaló inequality. Math. Z. 256(2), 379–395 (2007)
Jiménez, C.H., Naszodi, M.: On the extremal distance between two convex bodies. Israel J. Math. 183, 103–115 (2011)
John, F.: Extremum problems with inequalities as subsidiary conditions. Studies and Essays Presented to R. Courant on his 60th Birthday. Interscience (New York, 1948), pp. 187–204, 1948
Klartag, B., Milman, V.D.: Geometry of log-concave functions and measures. Geom. Dedic. 112(1), 169–182 (2005)
Leichtweiss, K.: Über die affine Exzentrizität konvexer Körper. Arch. Math. 10, 187–199 (1959)
Mazj́a, W.G.: Sobolev Spaces, Springer Series in Soviet Mathematics, vol. 145. Springer, Berlin (1985)
Palmon, O.: The only convex body with extremal distance from the ball is the simplex. Israel J. Math. 80(3), 337–349 (1992)
Petty, C.M.: Isoperimetric problems. In: Proceedings of the Conference on Convexity and Combinatorial Geometry, pp. 26–41. University of Oklahoma, Norman (1971)
Schechtman, G., Schmuckenschläger, M.: A concentration inequality for harmonic measures on the sphere. GAFA Semin. Oper. Theory Adv. Appl. 77, 255–274 (1995)
Schmuckensläger, M.: An Extremal Property of the Regular Simplex. Convex Geometric Analysis, vol. 34. MSRI Publications, Berkeley (1998)
Schneider, R.: Simplices, Educational Talks in the Research Semester on Geometric Methods in Analysis and Probability. Erwin Schrödinger Institute, Viena (2005)
Zhang, G.: The affine Sobolev inequality. J. Differ. Geom. 53, 183–202 (1999)
Acknowledgements
The first named author is partially supported by Spanish Grants MTM2013-42105-P, MTM2016-77710-P Projects and by IUMA. The second named author is partially supported by Fundación Séneca, Science and Technology Agency of the Región de Murcia, through the Programa de Formación Postdoctoral de Personal Investigador, Project reference 19769/PD/15, and the Programme in Support of Excellence Groups of the Región de Murcia, Spain, Project reference 19901/GERM/15. The third named author is supported by CAPES and IMPA. The second, third, and fourth authors are partially supported by MINECO Project reference MTM2015-63699-P, Spain.
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Alonso-Gutiérrez, D., Merino, B.G., Jiménez, C.H. et al. John’s Ellipsoid and the Integral Ratio of a Log-Concave Function. J Geom Anal 28, 1182–1201 (2018). https://doi.org/10.1007/s12220-017-9858-4
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DOI: https://doi.org/10.1007/s12220-017-9858-4